Overnight camping isn't allowed in the national parks except in campgrounds, but free "boondocking" is allowed on U.S. Forest Service land. I don't believe the
Bridger-Teton National Forest is far from Jackson, so you might look into
dispersed camping there. Some restrictions apply -- I think you must be 100 feet off established roads but within 300 feet, and at least 200 feet from lakes or streams, etc. (I'm not an authority on it; just going by things I've read in the past.)
It might be best to contact the Jackson Ranger District:
25 ROSENCRANS LANE
PO BOX 1689
JACKSON, WY 83001
Office hours Monday-Friday; 8:00 am to 4:30 pm Phone: 307-739-5400
Email: Not available