Wishin,
I too am planning to be in Yellowstone, Grand Tetons in June. Belonging to a Women's RV group, I have had a lot of insight from members who have been there done that as have the folks on here. More that one has told me that Fishing Bridges sites are very tight. Not much room to maneuver. That being said, I booked reservations at Grizzly RV park in West Yellowstone and am going to do my sightseeing from there and then move down to Teton Valley Park to do Grand Tetons. Fortunate to have one member of group as a native of Jackson, and is going to show me the Tetons up close. You might want to do some checking on campground in that area. My thoughts are that even if it's a bit more expensive, it's worth it. I will be coming in from Livingston and down to West Yellowstone by way of Bozeman. Am originally starting from Michigan Happy Travels.
