I've never been in June but I go Presidents week all the time. We got there on a tuesday last year and it wasn't too bad (however, presidents day weekend is bad)but the crowds tend to disappear. You may want to go from Wednesday of Presidents week until tuesday of the next week. On Friday MGM was empty at night (Epcot was a zoo)but on Saturday, Epcot was empty.
The best thing to do is get to the parks early and use FP.
We just got back going the week before Thanksgiving and truthfully, I thought the Monday, tuesday and Wednesday before Thanksgiving was worst than Presidents week. We walk on TT -twice and then rode soarin(w/FP) on Saturday morning (the tailend)of Pres Week and the day before Thanksgiving TT was a 90 min wait at 10:00.
Have a great trip. We are going the week before- Sat Feb 10 -16 this time.