This hike leads you into the Absaroka-Beartooth Wilderness, in the Absaroka mountains.
I am posting this because the other Elephant Head Mountain post is somewhat misleading - we weren't able to find the route from the backside approach, through Suce Creek. This should give a good overview for both routes.
Elephant Head is a pretty easy mountain to summit. Fairly short hike in and an easy hike out.
By the way, I should mention that I am not sure my coordinates are correct - I do not have my map at the moment and tried to find it on Google Maps, but it's not labeled, so don't go by them! Additionally, I therefore don't have the exact elevation of the peak. This should be helpful anyway.
To access via Mission Creek trailhead, use the eastern most exit for Livingston (Frontage Rd) and merge onto Frontage Rd. Look for Swingley Rd on your left. Drive about 10-15 minutes until you hit Bruffey Ln, on the right after a really windy uphill ascent. Impossible to miss, it's just about the only rd you can turn off onto. Drive until you see the sign for 360 Ranch. Turn into "their" driveway and continue through all the gates, LEAVING THEM AS YOU FOUND THEM until you hit the trailhead.
Please be sure to be respectful of the landowners, their horses, and guests - they are already unhappy about the land use easement through their property - or so I hear.
To access via Suce Creek trailhead (what I did - highly recommended, but you'll have to have a pick-up or shuttle car.) travel south of Livingston into Paradise Valley and turn left onto East River Rd. It's the first National Forest Service access on the left, maybe 2 or 3 miles? Finding the trailhead is easy, and once on the trail, it forks pretty quickly and you want to go right - bring a map!
Suce Creek trail was great - no bear scat and pretty easy mostly, with a few more difficult steep sections - we were backpacking. Once you get to the first saddle, you're in the W Boulder Divide drainage. Head to the next saddle to your right and that is Elephant Head. You must descend until you can see the easy ascent. There is no way to the peak from this backside unless you're going to rope up. I wouldn't trust rope on this sharp seaside limestone.
From the peak, you do have cell service, (at least my Verizon iPhone did) you can easily call someone to meet you at the Mission Creek trailhead if you didn't want to shuttle.
I DO NOT RECOMMEND trying to do a loop to the Livingston Peak trailhead. The East Baldy Basin trail was filled with fresh bear scat, horribly overgrown and unmaintained, very buggy, and not pleasant. If you're up for an adventure and to test yourself though, then I would recommend it. If you did Suce Creek to Mission Creek, it would be doable, easy, and enjoyable as a backpacking trip.
If you're camping, make sure you camp far enough from streams to meet National Forest Service requirements. I do believe this is wilderness area, so no mountain bikers - hikers and equestrians only.
There are few flat spots in the basin near the top, but there are some. Plenty of trees for hammock camping, fewer flat spots. Few big trees to hang a bear bag from - so you'll have to utilize two smaller ones or bring a bear can.
Plenty of water in July when I went - I have a hard time imagining there not being water. You won't find it on the peak or on the last ascent, but you'll always be within, at the most, a mile of water.
There are none - I've done a lot of internet searches for this mountain and there is limited info out there, thus I created this page, despite some incomplete info.
Climbing the Peak
At the worst, there might be a bit of scrambling, but I must say I didn't actually go to the top. The way to the top is pretty darn obvious from the Mission Creek trailhead, and visibly doable without ropes. It's probably not even a class 4 scramble - but again, I wasn't up there. You will have to go off trail to get to the peak. I didn't approach from that direction, so you'll have to use your own judgement - and if you can't figure it out, you shouldn't be up there in the first place! It's that easy.