We did 19th June - 4th July last year and there was a huge difference between the first week and the second week. Magic Kingdom became virtually impossible in the days leading up to 4th July. We got round it by making an early start and leaving by lunchtime. The earlier week of the holiday, crowds were moderate, but managable. I wouldnt have though you would have too much trouble in early June. Be prepared for the worst by finding out Early Entry days (avoid unless you are an onsite guest) and go all out for early starts. Dont be too ambitious...the parks wont go anywhere...dont be afraid to LEAVE if its too hot / too busy and return later in the evening or another day!!Sometimes moving to a different park is a good strategy...Epcot is a great choice for the afternoon. It never seems busy at that time, even in peak periods.
Its easy to get up at the crack of dawn in the early part of the holiday due to the time difference. Aim to blitz your "must do" attractions then, leaving the other stuff (parades, shows, shopping etc) till later in holiday.
Have fun!
Allie