Yes, another Elk Mountain. This one is in the north section of the Wyoming Range, just south of the Grand Canyon of the Snake River. Very near the highpoint of this long ridge of a mountain (running roughly north to south) is the unused Elk Mountain Fire Lookout. There is a trail all the way to the Lookout. From the summit ridge, a section of the Snake River is visible to the northeast.
Easiest access is from the town of Alpine, Wyoming (almost on the border with Idaho). Alpine is reached from Highway 89 from the south or the Grand Canyon of the Snake River, or from the north via Highway 26. The Greys River and the Snake River join in the vicinity of Alpine. Look for a well-signed Forest Road #10138 heading south and east from town (this is the Greys River road). Follow this unpaved but good road parallel to the Greys River until it branches north (away from the main river) on the Little Greys River road 10124. About 1 1/4 mile from this road intersection is a short spur road - Trail Creek road #10182. Start the hike here.
Follow Trail Creek trail #069 as it climbs out of Trail Creek drainage in a north and east direction. The trail will branch on the summit ridge just south of a prominent wooded knob. There is no signage here - take the southerly branch. The main trail will continue north and east to the Snake River. Follow the southerly branch on the wide open mountain ridge - the Elk Mountain Fire Lookout will soon come into view. Expansive views open into the Greys River valley, the Snake River, and even the Tetons are visible to the north. There are a a couple benchmarks in the vicinity of the lookout shelter. The hike (one way) took me about 1.5 hour at a moderate pace.