Mountainous Idaho in the northeast of the States hosts 325 miles of the Continental Divide Trail, parts of the Great Western Trail and parts of the Pacific Northwest Trail.
All these trails are of great length. But if this is not enough, you can venture on the 1200 miles pure Idaho Centennial Trail, running north-south along the crests of the Rocky Mountains with its many lakes.
Targhee National Forest borders on Yellowstone National Park (Wyoming). There is very good walking in Nez Perce National Forest, Idaho Panhandle National Forest, Sawtooth National Forest, Challis National Forest, Payette National Forest, Targhee National Forest, Salmon National Forest and others.