An improved loop would be to start at Kearsarge Pass and come out via Baxter Pass.
There are two issues with Baxter Pass, (1) it's very rugged, so I wouldn't recommend it for the inexperienced; and (2) you will need to hitchhike or arrange a shuttle. Park your car at the Baxter Pass trailhead early in the morning, hitchhike (it may be a long while before a car comes) to the Kearsarge Pass trailhead, and start your walking.
The best side trips along the way are Dragon Lake (easy) and Sixty Lakes Basin. Sixty Lakes Basin may be worth a couple of days. The north end of that basin, where it dead-ends at a cliff, has an incredible view. The southwest part of the basin has an easy x-country pass into Gardiner Basin, another place to explore if you are experienced at x-country.
Kearsarge-Baxter is one of the best loops in the whole Sierra. With its eastside entry and exit, and quicker access to the high country, it is much superior to the traditional loop that starts at Roads End to the west. You avoid all of the walking-a-dusty-forest-trail parts. The passes themselves, Kearsarge & Baxter, are both very memorable and pretty.
If you are a strong hiker and like to go lots of miles per day, you could continue hiking north along the JMT and exit at Sawmill or Taboose Pass instead of Baxter Pass. Sawmill and Taboose, like Baxter, are long, rugged and remote. If you make it all the way north to Taboose Pass area, you will want to do a side-trip to the famous Bench Lake and maybe Arrow Peak.