If you can spare 1.5 hours, you can expand up to the Mirror Lake Highway, were again, you'll find many campsites already taken at established campgrounds on weekends, but you could probably find something.

I camped at the Spruces in Big Cottonwood last fall and it was very nice with lots of moose (who seem to be permanent residents).

If you are more adventurous, you can scout on Google Earth for free camping out on the Forest Service roads in the Kamas area but most obvious spots will be taken up by the permanent RV'er types who lock them up all summer for their extended family caravans.