We go to South Dakota every spring and stay in Custer for two weeks for great hiking. We make our reservations as soon as the open, a year ahead of time. They fill up quickly, but sometimes you can get lucky if you catch a cancellation just as it shows up. We did that on a trip to the west a few years ago when were heading to the the Rockies around Boulder and west to the coast. We were going to stay in Wind Cave National park which is just short of Custer, but was watching the reservation site just in case. I don't know if they added a wait list and notification list like some states have, but I do know their new a few year ago reservation site is one of the worst I have ever used.
Hiking Bear Butte is also very nice and we have done it several times. It is Rocky and steep with a lot of loose rock to go across but really nice at the top with the view. It is not easy hike, though.
The Badlands are beautiful and we stayed in the National Park there a couple of times. It has now gotten very hard to get in as it busy now and the last time we were there they were so disorganized they didn't know if they would have open sites that day and all we could do was go to the campground and ask everyone if they were leaving that day. There is a very nice KOA that isn't the typical parking lot you usually see just south of the Badlands on the road to them. There is also a private campground on the road to the badlands to them from I90. It doesn't look stellar, though, so we have never tried there.
Usually now we just go there for a day and drive around, maybe hike some nice but barren with big rock formations trails and then go stay elsewhere.