If you have your Madison campground reservation already that's all you'll need for May. (For others questioning if it's open, Madison opens April 29)
I hope you plan to visit Grand Teton Nat'l Park, also. It's right next door but I'd highly recommend that you move down to that park to stay and do things there. We live Gros Ventre campground which has 300 non-reservable sites and rarely fills and definitely not in May. There's also Colter Bay campground which also has 300 non-reservable sites. If you're ready for full hookups after Madison, I'd suggest Colter Bay RV in the Tetons. It's right next to the campground of the same name.
https://www.nps.gov/grte/index.htmHave a great trip! It should be beautiful there in May and you'll see many young critters..... and don't make any more reservations. :)