It might be a problem, especially if there are a lot of tents set up near the few trees suitable for hammocks. Here is a short video of the area - taken during the off-season (now imagine the area crowded with tents):
https://www.youtube.com/watch?v=e0gUsZ21tesI know this might set off a long thread of hammock vs. tent discussions, but leave the hammock at home. Use a tent or, if the weather is nice, just use your sleeping bag - the mosquitoes are just about gone now and you can watch the stars.
Whitney frequent Whitney hikers in this forum might give you a better idea about trees, etc., at Outpost. What do you say, Steve?