Hiking opportunities abound in Yellowstone National Park as well as the immediate area of the cabin. The Henrys Lake Mountains to the east of the cabin contain the impressive Sheep Mountain (10,200 ft.), Coffin Mountain (9,967 ft.), and Targhee Peak (10,300 ft.). The Sheep Creek trailhead (elev. ~ 6400 ft.) starts just one mile from the cabin and climbs five miles before reaching Sheep Lake, at an elevation of about 10,000 ft. The trail continues through the Henrys Lake Mountains and connects with the Continental Divide Trail and with Mile Creek Trail, which eventually returns to the valley floor. The entire trail, from Sheep Creek trailhead to Mile Creek trailhead is about 25 miles. The Targhee Creek Trail is another great trail in the area. The trailhead is on the way to West Yellowstone, about 19 miles from the cabin. All of these great trails are filled with bears, so come prepared.
A number of websites give excellent trail descriptions of the Sheep Creek, Mile Creek, and Targhee Creek Trails. Many are described from the perspective of mountain biking, but provide excellent photos of the trail and give you a glimpse of how scenic the area is.