Is it hard to find a good hammock site in the Adirondack high peaks if not staying at a designated campsite?

I am looking into backcountry camping in the Adirondack high peaks in early to mid September. Plan is to hike in with my gear to a basecamp site and and stay a couple of nights. I plan on day hiking some 46ers with a day pack during the day. This will be my first time backcountry camping in the Adirondacks. The the backcountry camping places I usually hike, I am able to reserve my specific site in advance. I am a bit nervous that when I arrive at the non-reservable, first come first serve primitive site/lean-to site I intend to stay at that it will be full. This would force me to hike to a site further along....that might also be full. To avoid this, I am thinking of bringing my hammock (Warbonnet Ridgerunner, Thunderfly tarp) so I can set up camp at a dispersed/at-large site (following rules for dispersed camping...not within 150 ft of trail/water, below 3500 ft, etc.).

I was just wondering how easy it is to find a decent site to set up my hammock at a place that is not a formal designated site? I've hammock camped in areas in the past where finding two trees is pretty easy but the thick underbrush or densely packed smaller trees made if difficult to string up the hammock and tarp. I'm planning to camp mid-week so that might make chances of securing a designated site better. I also have no preference on hiking in a specific area of the high peaks, so if there is an area that does not see as much foot traffic, or an area that makes it easier to set up a hammock, I am open to that. I appreciate any help anyone might be able to offer.