Mount Muir was named after John Muir, a famous geologist and founder of the Sierra Club. Being 0.95 miles from the famous Mount Whitney, most people probably don't even know that it's there. But the peak itself towers at 14,018 feet and can be easily acessed from the Trail Crest when you make your way to the summit of Mount Whitney. There isn't any sign that indicates which peak is Mount Muir from the trail, so make sure you study the map ahead of time.
You'll want to follow the route toward Mount Whitney that begins at Whitney Portal. The East Butress Route starts from the Trail Camp where most people camp for the night before heading to the summit of Mount Whitney. This route has no maintained trail and requires quite a bit of route finding. You start by following the trail toward the 99 switchbacks from the camp; leave the trail to the right after you gain a little bit of altitude past the lake. There are a couple of boulder hills to pass before you reach the base of the route. There might be some snow fields in late summer, but these can be avoided. Once you reach the base of the mountain, the route finding begins.
This route requires trad gear, rope, and a helmet. The rocks are pretty loose and easy to kick down, so make sure to be cautious when you step and avoid raining rocks on your belayer.
After a long weave around the mountain and past the first tower nearing the summit, the climbing gets easier and the terrain mellows out to a scramble. The summit area is pretty small and generally windy. Enjoy the view before scrambling down toward the main trail, and follow the trail back toward 99 switchbacks to return to the camp.