I usually take it easy the first few days, coming from near sea level, flying into Jackson. I usually spend the first night in GTNP or camp near Pinedale or Dubois. The first full day, I usually fish the upper Green River first or swing over to Dubois to fish the Wind. Short climbs up steep banks certainly causes me to be a little dizzy, so I play it safe. 2-3 nights seems to be enough to start a hike to alpine areas. Dayhikes near Pinedale to anything scenic are few, as has been noted. Clear Ck natural bridge, a walk along New Fork Lakes, possibly to the canyon are possibilities. There will not be solitude until you backpack and get off the main highways to the Cirque and/or Titcomb.
There's a photo someplace I saw of Elkhart TH in late August. Completely full. If you have to use it, try early morning during the week. One reason I always go after Labor Day. This year I'm forced to go out a week earlier but will start in from TH's near Lander which are somewhat less popular. The week of Labor Day I'm leading a group in via Elkhart, but I hope the crowds will have thinned after the holiday weekend.