I posted this trip in a few places and received a few PM comments and questions, so to be most economical and informative I will relate here my thoughts and answer those questions about the trip and my recommendations.
My original plan was to enter Pants Pass, then go south over either Lawson Pass or Picket Guard Pass, climb Picket-Guard for a better view looking west, spend time at the last most eastern Picket Lake, go up Kaweah Basin and over Pyra-Col Pass, bag Eagle Scout Peak, and then return on the HST. Up to the last minute I was considering doing the same except to switch it up and go counter-clockwise. I still think that this works well as a loop trip.
I instead went over Pants Pass, then Picket Guard Pass, walked up the Kaweah Basin, then returned and ended my trip to the Basin by exiting east from the last Picket Lake to the Kern River and then returning by going the long way around on the HST. Yes, this exit was easy, and it should be clear enough from the map I published. I took the Picket Pass instead of Lawson Pass because of lingering snow, and it appears that both Lawson and Pyra-Col will be more likely to have late-season snow later into the summer, so this is something the June hikers should consider.
An alternative exit suggested to me is to exit going east from the last Picket Lake along its drainage creek all the way to Rockslide Lake. This suggestion apparently involves Class 3, whereas the route I took was almost all Class 2. An alternative entrance suggested is to come in the from the west, summit Lawson Peak, go down the ridge in the east direction, and then drop over Lawson into the Picket Basin.
I did not care much for much the basin to the south of the Kern-Kaweah River; this is the one you drop into from Pants Pass. The middle Basin, with Picket Creek, looked OK, but I did not walk up it and cannot offer an informed opinion. I can however emphatically insist that the most important destination should be the lower Kaweah Basin. Yes, that last Picket Lake is very nice, but it is the lower Basin that is exceptional.
If someone intends to just walk through east-west (or versa), like on a version of a modified HST trip, I recommend Pyra-Queen pass, (although I have not taken it), walk through the Kaweah Basin, and go out one of the alternate ways just mentioned. Hopes this helps.