We stayed at Stockade Lake South 2 summers ago and loved it. It is one of their smaller campgrounds so you might be limited length wise, but it worked perfectly for us. We were in site #16. The bath/shower facilities were kept clean, it was just a few miles from Custer so easy to run into town for provisions.
Wife said last night she'd like to go back this year, but she'd have to start preparing mentally now for the long drive from Texas....but everything is.
There is an incredible amount of things to see/do so you really won't scratch the surface with just being there a couple of days. We were there a week and still left enough to see to warrant a return trip.
If you're traveling with kids, you can easily spend an entire day at Evans Plunge Mineral Springs swimming pool in Hot Springs SD.
We were in Yellowstone/JH a few years back as well. Yellowstone was cool for what it is, but we REALLY enjoyed the Tetons and JH area ALOT more. If you can swing it, I recommend taking a horseback trail ride. Try finding one that is at least 3 hours minimum, the longer the better.
Also take the ferry at Jenny Lake and hike up to Inspiration Point, see the barns on Mormon Row, walk the downtown square, etc.
In all actuality, either place could be considered a sole destination trip, and trying to split up one trip into two awesome places will be doing a disservice to both of them.