June is very, very hot. Mid-to-high 90's and very humid. On top of that, there are afternoon pop-up rainstorms almost every day. I would definitely insist on a break after lunch until 4 or 5 pm. Even if you're having a great time, most kids are melting in that weather and are glad to get on the nice, cool monorail or bus and go back to the hotel pool for a while. We are going this month but taking the after-lunch break; DS3 needs a daily nap and DH and I want some downtime, too.
we went to DHS and left a little after noon, we have never done this before but for some reason it was really hot that day and we were all miserable with the heat so we left and went swimming (we got FP for TOT and RR before we left so we would be able to rode when we got back) came back late afternoon in time for the fp window to ride did a few other things we had missed and watched Fantasmic
But at the same time, he can stay up quite late when we are active. If we didn't push it, my other kids get up at 6:30 for school and could easily staty up to 10-11pm.