This year, at the end of June, the best day-hiking may be in Yosemite Valley. Honestly, I think at peak water flow, there are more spectacular sights in the Valley and hikes from the Valley, than anything in the Tuolumne region. A hike up the Merced, camping between Little Yosemite Valley and the Merced HS camps (or at the campground at the Merced HSC) is a classic waterfall adventure. Yes, it can get crowded until you pass Little Yosemite Valley. Or even stay at Camp4 and day-hike from there. For example, Four Mile Trail- Panorama Trail, back via Mist Trail and JMT to Happy Isles. If doing the Merced hike, you can even make it a loop by cutting over to the JMT from Echo Valley on your return. Maybe you have already done all the hikes in the Valley, and in that case, I would vote for Glen Aulin, simply for the waterfalls. But even though I have done the Merced River hike a million times, it is my favorite every year at peak runoff. I never get bored with this hike.
Another thought- probably not likely, but Tioga Pass may not even be open to overnight parking.