Not sure from your post if you're RV'ing, car camping, motorcycle camping, horseback camping, or want to pack in to a wilderness area ....
Given the millions of acres of Nat'l Forest and Wilderness lands upon the doorstep of Dubois, your choices are phenominal. I can't imagine how a KOA there survives except for those who are too lazy to get out of their RV for more than a few steps ...
You are right between the lands of the Shoshone National Forest/Wilderness, and the Bridger-Teton ranges at Dubois, and ... of course, not too far out of Yellowstone. There are trailheads into the forest lands that start as close as 3.5 miles from Dubois, and many in the 10-15-20-25 -30 mile distance from town. You'll be heading into some "isolated" and "remote" country, for sure. Some trails are accessible by car, some only by a 4x4 off-road equipped vehicle, some only by horse or hiking in.
Be advised that this is active bear country, and the "bear proof" food storage/attractants regulations are now in effect. Best to contact the National Forest folks to find out what's required ... the Wind River Ranger District is based in Dubois, and they can help you. Or visit some of the websites ...
www.fs.fedus/r2/shoshone/ ... for example. There's also some other private fee based camping areas around Dubois, you can google for camping in the area and come up with their sites.