I like millonas' idea, of starting with the Grand Canyon of the Tuolumne, and combining it with the Tuolumne-Reds Meadow section of the JMT. Very creative, and an improvement over starting from Happy Isles.
The downside is that you'd be doing that route in a low-water year, and too late in the season, to boot. The whole point of going down the GCT is to see the waterfalls when they are blasting at high water, which is generally late June of a high-snowpack year. This year you will get more of a trickle, especially with an August start.
I'd suggest buying the "Sierra South" book and doing the North Lake to South Lake loop. It's the best part of the JMT, and the sections that lead you to and from the JMT are just as breathtaking. The hitchhike is very short and easy to organize, compared to most of your other options in the Sierra.
If the mileage of the official loop isn't enough for you, you can pad it with side trips to stunning locations in that area. In fact, a lot of people do this loop just to get at the side-trip destinations! If you're comfortable with a little cross-country hiking (sometimes assisted by "use trails"), these side trips will provide somewhat less trampled camping spots, too. Assuming you're going from North to South Lake, some of the side trips I would consider are:
– Muriel and Goethe Lakes (camp at Muriel, to which there is an easily followed trail)
– Desolation Lakes (easy-to-follow trail; camp at Upper Desolation)
– Darwin Lakes (follow the use-trail there) (camp)
– The little basin behind Sapphire Lake (no trail; rocky; not to camp just to explore)
– Davis Lakes (no trail; explore or camp)
– Ionian Basin (no trail; rugged and for very experienced backpackers only)
– Dusy Basin exploration (camp anywhere)
– Barrett Lakes/Palisade Basin (a x-country hike from Dusy Basin) (campable)
– Chocolate or Ruwau Lake (trailed; near South Lake trailhead if you happen to start from South Lake and need a short first-day hike in)
Piute and Bishop Passes are among the first on the Inyo National Forest website to be snagged for permits. You need to get online and just reserve one, ASAP. The system allows you to change your reservation later. For various reasons, I'd suggest going north-to-south, so, starting at North Lake and going over Piute Pass.
If you plan to go off-trail ever in your trip, as I suggest above, be sure to choose "cross-country" as the type of permit you are getting. (It does not affect permit availability; just limits your group to 8.)
IMO the section southbound out of Vermillion to MTR is one of the hottest, driest, least interesting sections of the JMT. And this year, with the low snowpack, Lake Edison may be drained again, nixing your boat shuttle and not helping the scenery.
– Elizabeth