"Maybe out of Florence Lake or Courtright Reservoir?"
One problem is that those places are not in SEKI, and you would have to walk for a day or two just to get into SEKI. There is quite a number of trailheads that are a day or two outside of the parks, and only a few that are closer. Onion Valley is just a few hours of walk from Kings Canyon at Kearsarge Pass, but I would not call that area less inhabited.
SEKI has a web site with all of the major trails listed with descriptions. Then, you can check those against the trails with quotas that are already taken by reservation. Those are the most popular trails. Then you pick a trail that nobody else is picking.
If you drive to Cedar Grove, try hiking up the Lewis Creek Trail over Kennedy Pass and Granite Pass, and down the Copper Creek Trail to Roads End.