As I mentioned in my earlier post, we always stay at Gros Ventre campground when visiting the Tetons. Never have had any problem getting a site. Most recent was in 2014 and we spent 5 nights there during the last two weeks of June. I'm sure you will have no trouble getting a site there arriving early to mid afternoon, especially since you plan to arrive mid-June. As the month goes on, it gets more crowded, although there were still empty sites when we left on June 29th back in 2014.
We also spent a night at the Atherton Creek Forest Service campground on Slide Lake east of Kelly. A bit further from the action, but a nice place.
