Looks like Seek the Peak doesn't require hikers to use any particular trail. I'm guessing the entire mountain will be pretty crowded, but definitely stay away from both the Pinkham Notch trailhead (most any trail on the east side) and the Ammonoosuc Ravine trail (on the west side). Jewell Trail (up to Mt Clay from the Ammo trailhead) might be a tiny bit less crowded, likewise the Glen Boulder trail (via Boott Spur from its own trailhead). A really pleasant hike that should be long enough to keep the STP crowds down is via Crawford Path (above treeline from Mt Pierce, over Eisenhower etc) ; you should also have some solitude for most of your trip if you take the extremely long approaches via one of the big valleys, either Great Gulf (ie, from the north) or the Dry River Wilderness (from the south).
Caps Ridge is surprisingly crowded most of the time (after all, Jefferson has great views, no train or road, least elevation gain in the northern Presis), but the traverse from Jefferson is not a bad idea either.