Well, April scores for the 2016 cycle will probably be available late May; the only reason they're not being released in the usual 30-35 days this go-around is so that AAMC can use the April and May exam scores to create a standard score scale for the new exam.
However, getting your scores back in June is not such a bad thing, and you have two options. If you feel relatively confident about your score, you can submit your application June 1st, but only to one lower-tier school. Your application will take 3-4 weeks to get verified and then be sent off to schools late June/early July. So by the time you get your MCAT score, you'll be very close to if not already done with the verification process, and at that point you can either a) add on the additional schools you want to apply to that make sense based on your score, b) decide you need a retake and wait to add on additional schools until after your retake, or c) decide to reapply next cycle if you need to retake but don't want to risk your chances by applying super late this cycle. The reason why you'd want to only initially apply to one school you're not that interested in attending is so you won't be listed as a reapplicant next cycle, which *may* hurt your chances.
Your other option is to hold off on applying to any schools until you receive your MCAT score. If you, for example, get your score back on June 16th and you like your score, you could submit your application that day, and it would be sent off to schools mid-July. That's honestly not much of a delay at all.
