Just got back from an unplanned visit to Yellowstone 8/3 - 8/9 (headed north to escape the rain in Colorado that week)
Was very apprehensive, as it was my first August visit, and without any reservations, but called on 8/1, and got two nights at Madison on 8/3. From there, left Madison at 7am, headed to Norris (first come campground) and found a site for the rest of the stay.
There were at least 6 others hunting for spots while we setup camp. I wouldn't go hunting for a new spot any later than 7.
Bridge Bay and Fishing Bridge are fine, but too exposed for my taste.
Call and see what you can find first days. If there is nothing, call each day till you do.
Other options for the first couple days is Gros Ventre in Grand Teton. Not a good base camp, but gets you within striking distance. Or Lizard Creek, at the northern end of Grand Teton.
