This is difficult to know. Verification times through the AMCAS have become increasingly long. So by the time you get your scores, your AMCAS may just be recently verified (it depends on when you sent it). The longest wait time I have seen on the forums is 8 weeks for verification. However, it does not take long to add schools and for them to receive your primary after verification (around a week, not sure exactly).
So I would suggest taking the mcat when you are ready, even if it means taking it in late July. I hope this helps.