This is a long post with details that will help those interested in improving their chances for JMT permits (some of this also applies to reserving yosemite campgrounds, and anything else reserveable through recreation.gov.)
Asolthane's link in this thread provides general information about obtaining permits within the various agencies.
Here is some valuable, more detailed information about how the recreation.gov website processes requests and cancellations, and a proven strategy for obtaining difficult INYO "Exiting Whitney Portal (Trail Crest Exit)" permits.
Recreation.gov permits for INYO trailhead quotas become available for reservation at exactly 10:00 am EST 6 months in advance, and when a permit gets cancelled the quota becomes available for reservation again within 24 hours of cancellation (this becomes important later on--keep reading).
It is advisable to get on the website early so as to navigate through the process of selecting the trailhead, entrance date, etc. ahead of time, all the way until the page with the "book permit" button. If you click the button too early, you will get a message saying there isn't availability. If you continue to click "update availability," at 10:00 am you will see the quota become available, at which point you can select the available day (each day has a box in the calendar bar, indicating the quota availabile), then scroll down and click "book permit." Note: if you don't click anything for a while as you prepare/wait, the session will time out, so be sure to keep your session active by clicking "update availability." Note: if you merely refresh your browser, the dates you entered may revert back to something else so be careful to check that. Clicking "update availability" instead of refreshing the browser should keep your desired dates in place.
Unfortunately, for some limited or high-demand quotas, waiting to click on the day box in the calendar section when it actually becomes available, and then scrolling down to click "book permit," isn't fast enough! Note that there is a date field next to the availability calendar. As long as your desired date is showing in that date field, you don't have to wait to select the day box on the calendar. Just click "book permit" at 10:00 am. In fact, I've found that you can click "book permit" a half second early due to processing times. In a real test, I had 4 computers all vying for the same permit, and the only one that got it was the one I clicked a half second early.
Note: the INYO wilderness permit system clock is on official NIST time. If you go to time.gov you can watch the clock as 10:00 am EST approaches.
Another way to increase your chances is to use multiple recreation.gov accounts, one on each of multiple computers. Don't worry, this isn't shady dealing...in fact, the recreation.gov representative and the INYO wilderness permit office representative that I spoke with both recommended the strategy of having others in your party try for the same permit. Just be advised that only the permit holder or the persons designated as alternate group leaders (up to 3 per permit request) can pick up the permit, and must be on the trip. Note that nowhere on the INYO website does it say that the trip leader or alternate leaders have to be adults (though that may be assumed). They do need government ID, however. In my case, the other party members are my two minor children, so I registered separate recreation.gov accounts for each and named all of us as alternates.
Now, things get tricky when you want to exit Whitney Portal (trail crest exit), because in those cases, the EXIT DATE has to be within the 6 month reservation window. The problem is, by the time your exit date becomes available for reservation, the trailhead entrance quotas have already been reservable for a time (i.e., if you want to enter a TH on Aug 15 and exit whitney portal on Aug 30, by the time you can try to reserve a permit with whitney portal exit (Feb 30), the entrance TH has been reservable since Feb 15, and the entrance quota is likely to be booked already by others who aren't exiting whitney portal but who are entering that trailhead on the date you wanted.)
One option is to select a different entrance trailhead, or a different entrance date (Note that not all INYO trailheads access the JMT. See the detailed descriptions of each TH on the recreation.gov website or the INYO website so you know which THs are real options for connecting to the JMT.) If you have TH or entrance date flexibility, its good to have those additional options. If you don't have that flexibility, then here is a proven strategy:
1) Using the reservation strategies above, book the entrance TH and date that you want as soon as it becomes available (10 am EST 6 months before), but "visiting" mount whitney instead of "exiting" mount whitney (trail crest exit). You can exit cottonwood lakes or some other trailhead. Book multiple permits (separate recreation.gov accounts) if desired, because you then...
2) Cancel this permit 24 hrs before your desired whitney portal exit becomes available for reservation (assuming the entrance quota on your desired TH is filled). This will have the effect of making the entrance quota become available for reservation again, in time for your attempt at reserving a trail crest exit.
NOTE: there are non-refundable reservation and cancellation fees. You'll need to decide how many multiple permits to try for (so you can cancel them so as to make your desired entrance TH available when your whitney portal exit date becomes available for reservation (within the 6 month reservation window)). For my situation, it was worth the extra ~50 bucks in fees I incurred to figure out how the system worked and to pretty much guarantee that I would get what I wanted.
NOTE: when a permit gets cancelled on recreation.gov, the system determines a random time within the next 24 hours to make those spots become available for reservation again (I learned this by talking with the recreation.gov and INYO wilderness permit office reps--they both told me the same thing). In my experience learning the system and obtaining permits, the spots became available for reservation again as early as 15 minutes after cancellation, or as late as 20 hours after cancellation.
NOTE: cancelling a permit does carry a risk that someone else will notice and reserve the entrance availability before your exit date becomes reservable. You'll have to decide how far in advance (15 min to 24 hrs) you want to cancel.
NOTE: Don't think you can "hold" availability by putting an available entrance in your cart (which starts a 15 min timer) and then "abandoning" your cart just before trying for an exit permit, because when you abandon the cart, the item does not become reservable again until the cart timer runs out (even after you abandon the cart).
3) have the permit ready as before, but with both the entrance TH and date, and Whitney Portal (trail crest) exit and date. Don't worry that the exit date availability calendar doesn't appear. As long as your date is showing in the date field, you can just click "book permit" at 10:00 am EST. NOTE that if your session times out, or if you refresh the page with the browser rather than clicking "update availability" the exit date may revert back to something else. Be sure the exit date stays as you want it.
4) click "book permit" (multiple accounts as desired for increased probability) right at 10:00 am EST (or even a half second early).
Additional information that is helpful in this process:
Whitney portal (trail crest) exit dates are given some leeway in practice, especially for longer treks, because they understand that someone might be delayed due to weather, or may arrive at trail crest a day early, etc. In speaking with the INYO wilderness permit representative, I learned that the grace period is "at least" a day before or after the exit date on the permit. Note that this means you could make multiple attempts for the exit date (i.e., one attempt representing a "day early"...the day before you think you would exit, another for the "sweet spot" date you think you will exit, and another for a day after.). In discussion with other JMTers and sierra guides, I believe the park service is very understanding and even lenient in this regard.
Note: you can't reserve conflicting permits on the same recreation.gov account. The system will return an error saying you have a conflicting reservation. One representative thought that the reservation won't even process to the cart (where it would normally be "held" for 15 minutes (with a countdown timer displayed), so trying to reserve a conflicting permit with the idea of "cancelling" a former less desirable permit (using a separate browser tab or whatever) while the timer counts down, is not advisable (or even feasible). It is good to know, however that a cancellation does take effect "immediately" on the account (even though the cancelled spots don't become reservable immediately but randomly during the next 24 hours).
Firefox seems to have trouble with the recreation.gov website. It is advisable to use Chrome. What I noticed is that firefox doesn't "update availability" very well (or at all), and even though it doesn't give you an error, when you go to click "book permit" it will return an error or not process the request.
Note that at high demand times (i.e., 10:00 am EST) the recreation.gov website may "crash" or not be able to process the request. Instead, you will get a page saying "oops, we are having difficulty processing your request" (or something like that). It will present you with options to "go back and try again" or to start at the home page. The fastest way is to go back and try again, but note that you will definitely need to re-enter your entrance and/or exit date. I found that if I clicked "back" in the browser that I ended up having to go further back. If this high-anxiety situation happens to you, as it did to me, don't lose heart. Keep trying. One time after this happened, the availability was gobbled up by the time I could click "book permit" again. However, I kept clicking "update availability" and after a couple of minutes, the exit quota showed 11 (of 15) available, so I was still able to get what I wanted, even though the 15 daily trail crest spots are usually gone by 10:00:00.5 (seriously!).
Also, its worth checking later in the day. One permit I obtained with a trail crest exit I found available at 6pm, after they were gone within a second after 10 am the same day. Someone must have cancelled a reservation or abandoned their cart, and the system made those spots available again later in the day! This takes luck, but its worth checking periodically.
Using the strategies above, I booked a few entrance permits (visiting, not exiting Mt. Whitney). Then I cancelled one ahead of attempting to reserve my "day early" exit. I didn't cancel it early enough (not 24 hours before trying to make the trail crest exit reservation), so I grabbed a different TH (which still had availability) with trail crest exit (I was ready with both my desired entrance TH and the alternate, and when the time approached and the entrance wasn't showing available, I just went for the alternate entrance). Then I cancelled a second "extra" permit 24 hours before my "sweet spot" exit reservation attempt and was able to grab the TH I wanted on the day I wanted, with the trail crest exit on the day I wanted (yeah!).
An added strategy for obtaining a whitney portal (trail crest) exit is to also put in for a permit starting in Yosemite (or elsewhere). It is hypothetically possible also to hold one permit with your desired entrance "visiting" Mt. Whitney while also holding a separate permit (from any available entrance) while exiting Whitney Portal (trail crest exit), as long as your itinerary allows for you to pick up both permits. If both permits are picked up within the same agency, that may raise eyebrows, but you could have your "alternate" leader pick up one or the other. (Note that the INYO website doesn't specify that the alternate leader has to be an adult...that is implied but it is not stated, so hypothetically anyone with government ID could be the alternate leader).
Note that for yosemite-based permits, you must pick them up at one of their permitting locations either the day before or the day-of the entrance date, during business hours (and before 10 am when picking up day-of entrance permits, unless you call ahead and tell them to hold your permit for later pickup).
Note that for INYO permits, those can typically be picked up one or two days in advance (or before 10 am day-of, unless you call ahead and have them hold it for later pickup. They also have a night box they will leave your permit in for day-before pickup after hours, if you want to take that risk...sometimes they forget, or the permit gets stolen, etc.).
It is also worth calling the Yosemite permit office, after checking their TH availability on this page: https://www.nps.gov/yose/planyourvisit/ ... lheads.pdf
I was lucky enough to notice availability for July 3rd entrance after I accidentally missed the deadline for faxed-in permits. I called, someone picked up on the first ring, and I booked it (but I cancelled it later due to the likelihood of snow that early this year--Mammoth mountain ski resort plans to be open passed Independence day!?). Again, this takes some luck, but its worth checking.
In my case, I'm starting NOBO on the Shadow lake trail, to Happy Isles, then starting SOBO from Duck Pass. This allowed me to also put in for a 4-day attempt at a SOBO permit out of yosemite, because I could pick it up as I pass through Tuolumne for same-day entrance or next-day entrance, or when I arrive in happy isles for same or next day entrance as well (even though I was only trying for that permit so I would have something allowing us to exit Whitney Portal).
As it turns out, I got the Yosemite SOBO permit on my 4th day of the 4-day attempt! Of course, I had to put in for that before my exit date became available for INYO reservations, so I ended releasing my Yosemite permit.
I hope this info is helpful.