I will second riding Teton pass. just park at the top of the pass and start hiking. to the south of the road is an area called edelweiss? Its a quick boot to the top. Im sure its tracked out by now but it should be avy safe by now, being a week since the last snow.
On the Jackson side I always camp on fs land at shadow mountain. its on the east side of 89/26/191 off of antelope flats road, just past the moose jct entrance to the park. There are also a places to camp in snake river canyon south of Jackson.
The road to Targhee passes through NFS land so there are places to camp there. Ive renegade camped by just pulling off the road and pitching a tent after dark and leaving early AM.
Google for specifics.
Its gonna be COLD this time of year so the lower you camp the better.
And also, Teton Pass is the steepest paved road ive ever driven on so if your vehicle's engine is prone to blowing up, be careful.
North of the park is towgatee pass. Which is F'n beautiful. I have never ridden there but the pass is at 10K so Im sure you could find some easy access to snow.