It would help to know where you'd like to do this loop. I'm assuming, therefore, somewhere in the U.S. May can be a tricky month depending on how much snow (if any) an area got over the winter. And whether the trails suffered over-winter damage (fallen trees, slides, etc.) can also be a factor. As can temperature in that some of the desert areas are already starting to heat up by then. But here are some ideas: (1) Rogue River Trail (bootsonthetrail.blog/2015/0…), 40 miles with a shuttle or you can do an out & back from either end (2) Wild Rogue Loop (bootsonthetrail.blog/2017/0…), 30 mile loop (3) Buckskin Gulch/Paria River Loop in Utah (bootsonthetrail.blog/2018/0…), around 30 miles but permits may be an issue (4) Upper Paria River in Utah (bootsonthetrail.blog/2016/0…), a 26 mile loop, no permits required I'm sure there are plenty of other possible loops in the Midwest and East. Maybe you'll hear from other's about those.
Thank you for the suggestions Bruce. I will keep them in mind for my next adventure. I've decided to do the Grand Teton Crest Trail last week of May. It is not a loop but I can take a shuttle from one end and meet my car on the other. I guess I can deal with some snow. I was looking at the average temperatures at that time and it says 60s during day and 40s during night.
The Teton Traverse looks like an amazing hike! There's a snow monitoring station (nwrfc.noaa.gov/snow/snowplo…) to the west of your route that you might want to keep an eye on with respect to how much snow you're likely to encounter in late May.