It's much more crowded at the very end of June compared to the week after the 4th. We used to go at this time and stay through the 4th, but switched to arriving around the 3rd and staying after (usually, when there aren't golf tournaments... The price I pay for having a fiance with a single digit handicap...). The crowds really build in anticipation of the 4th. If you use fastpass and take your time, you'll still have fun, though!