I am looking to climb in July just want to find some beta on routes. I am trying to avoid the more popular one's, We might be there on JULY 21st which is the STP hike. Definatly want to stay away from that area.
Try the Caps Ridge Trail. The trail starts on the Jefferson Notch Road and heads towards the summit of Mt. Jefferson. You can skip this summit (or not), and then head south to Washington. It's a nice hike because you get above treeline quickly. A little longer in length than some of the more direct trails, and a little more rugged, but the elevation gain is less because the Notch Road starts higher (and you shouldn't see the crowds you might elsewhere). Not sure of the exact distances -- most of the trail can been seen on Google Earth -- easy to map out.
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.
A less traveled and more challenging route to the summit of Mt. W is the Davis Path. It starts from a parking area near US 302. The distance from there to the summit is about 15 miles and the elevation gain is about 6750 feet.