Lots of Beautiful entrances from the Eastside.
the next one up is out of Independence via Onion Valley.
trailhead is 9,000ft. up over 12,000ft Kearsarge Pass to beautiful backcountry. can camp relatively soon at 10,000 or hike a bit more to get below that. this takes you south on JMT to Vidette Meadow and Forester Pass, or north to Rae Lakes area.
it is fairly easy to get a permit out of Onion Valley especially mid week. I think its 60-40 reservation and walkin.
if you want an out and back, one of the nicest places is out of Big Pine.
North or South fork of Big Pine Creek up to the Palisades. my preference is North fork because there are a couple nice lakes around 10,000 ft. Camping at 1st or 3rd Lake is very scenic.
North Palisade is the most alpine-like area of the Sierra. from your camp, you could day hike up to the Palisade Glacier (12,000ft), the biggest in the Sierra, though small by Northwest standards.
out of Rock creek is also nice.
out of Bishop is also nice.
out of Mammoth-Reds Meadow is nice, going north is best, to the Minarets or Thousand Island Lake area.
much of your Eastside backcountry camping will be at 10,000ft if you want to see the best places.
but there's always Diamox, works great for altitude.
how far do you want to hike ?