We tried to snag a BLM spot, but didn't make it in time.  Many of the campgrounds in the area book up fairly fast in the mornings.  However, we heard Gros Ventre normally has spots because of the size of the campground - something like 300 spots.  It's a big campground that could feel impersonal, but there are many spots facing the Tetons that feel a little more "personal".  It's really well located - super easy distance to Grand Teton National Park and the town of Jackson.   What it lacked in "character" was made up in ease of use and central location.  Also, perfectly located on the way towards Yellowstone if you are planning to do both parks in one trip.