We have always tried to go as early as possible to avoid the heat, but from the middle of June through August there isn't much difference temperature-wise. You are only looking at a difference of one degree, 91 to 92, for average highs, and we have experienced highs in the mid 90s many trips between late June and early July. June actually has the highest average precipitation.
The earlier in June, the better for both temps and crowds. Mid June through the first week in August will be the hottest and most crowded times of the summer.
This year we are visiting the first week of August do to other obligations, and I'm fully prepared for the heat. I can't imagine it will be any worse than the mid 90s that we have experienced on previous trips in June and July. I'm hoping we may get lucky that crowds will start to die down by then, as schools in the South start back. We are visiting friends in Atlanta the last week in July, and their son goes back to school the day we leave for Disney.