Hello! A lot of people like the “Beaten Path” in the Beartooths. Early season you could run into snow. However it is soooo beautiful! Lots of waterfalls and lots of options for distance and places to camp.
The Crazies are my favorite! I swear I’ve had outer body experiences in those mountains because I find them to be so enchanting! From the east side you can check out Twin Lakes from the Blue Moon campground, or any of the other lakes in that area. Even just driving into this range makes me want to cry in happiness! From the west side, my favorite place in the crazies is Campfire Lake, however it might be inaccessible that time of year due to a giant snowdrift that persists sometimes until August. This will depend on our snowpack this year. IF you are around when the high elevation snowpack has melted, you can to a traverse of the entire Crazy Mountain range (would require a shuttle though, or swapping keys with someone in the middle).
Probably the most popular trail near Livingston is Pine Creek Lake. You get to a great waterfall right away, and can continue up to the amazing lake for a 16 mile round trip. It is very popular though, I’m not sure I’d want to backpack there just bc of the amount of traffic. But it would be a great long day hike for you if you are in Livingston.
There are soooo many places in these areas! I’d suggest grabbing a map of the Beartooths and start dreaming!
Also, just driving the Beartooth Highway itself is so cool! It opens generally on Memorial Day weekend, weather permitting.