Early September is the best choice considering the criteria you have given. While it is true that there are less hikers in early June than later in the summer, there are reasons for this.
Black flies can be quite annoying then. June is also wetter and muddier than early September. Avoid Labor Day weekend, but there will be fewer hikers immediately after that. September weather tends to be dryer with more clear days than other parts of the summer.
Personally, my favorite time to hike in the White Mountains is from Labor Day until peak leaf season. That's in early to mid October, depending on elevation and weather specifics of that year.
Timing isn't the only way to avoid crowds. Most of the people will be on the AT, or doing day hikes on a few of the popular trails that afford a out-and-back to some nice vista or other destination. The White Mountains has a large network of trails, with most of them not crowded at all, even during peak season.
So don't do the Presi-traverse on the AT. You'll meet lots of people and have a hard time finding a place to camp at night. Instead, make a loop over more obscure side trails. It helps if you can spot a car, but there is also a shuttle bus system for hikers run by the AMC.
I've done two and three night loops in the middle of summer west of Crawford Notch and hardly run into anyone once I got past casual day-hike range from the trailhead. If you insist on going up to the Presidential ridge, especially near Mt Washington or Lake of the Clouds or Madison huts, you're going to run into people. But that's only a tiny fraction of the available trails, especially if you have the flexibility to not have to get back to a trailhead for three days.
Take a good look at the maps. Get the AMC maps printed on Tyvek. Those really do hold up well to water and other trail-abuse. The AMC does a decent job of keeping the trail information on those maps current. They are no doubt the best maps for hiking in the White Mountains.