Ok so just talking general rules here, obviously the White Mountains has unpredictable weather but from what I read . . .

Mid-May (Mother's Day) thru Mid-June (Father's Day) is usually the height of Black Fly season
May can also have snow/ice at higher altitudes and require microspikes.

September (after Labor Day) is the "best" time to backpack/hike since less bugs and crowds.

How is late June-late August? Any further breakdowns or "seasons" in this date range? Do you just have to deal with the crowds of people backpacking/hiking but otherwise usually pleasant?