Are you planning to stay at the High Sierra Camp there?
If not, on the way there there are a number of designated areas to camp
such as at Mehrten and Buck Creeks. You'll see why they have these
sites if you look at the topo.
If you stay at Bear Paw I would dayhike to the Redwood Meadow Grove
on one day and Hamilton Lakes on another.
Both of those areas are "must see" in my opinion. Either of those are
great spots to camp too. You can camp directly under a Sequoia...
how cool is that?
And to Hamilton and all the way to Kaweah Gap ... the scenery is only
second to Yosemite
There may be some snow on the trail but since it is southern exposure
by early July it shouldn't be too much.