You may want to also tie yourself to that tree!
As the Beatles sang, “You know, it just ain’t easy.”
What you are actually looking for is a 10ft by 10ft piece of ground that is level which really isn’t going to show up on a topo map, although, your odds may improve in what appears to be a flat area.
Of course, there are the trees. Trees in the Whites seem to love flat areas even more than campers.
Then, there are a few open areas. These are exposed granite slabs. This in turn requires a free-standing tent. But then again, the Presidentials is one of the last places that you want to be in a tent that isn’t nailed down!
Have you considered a hammock?
I make a habit of looking for a camping site early before dark (I don’t push myself on hikes anyway). In the Whites, I’m not very picky about a site because if it’s good enough for me to even consider, it’s probably the best one I will see all day!
For all these problems, it is such beautiful country that it is worth these hassles. I was just up there for a week in July and looking forward to another week planned in October. Good luck.