yes. many schools will send you secondaries within days of AMCAS submission, even though you've still got a while until verification. AMCAS must notify schools somehow. of course, it doesn't make any difference, since you won't be complete until you're verified anyway.
a June MCAT isn't too late by any means. but you do have to be a little better organized. the best thing to do is to submit AMCAS with one school listed (your state school, perhaps) that you know you'll be applying to, regardless. then you add more once the score comes in. this won't change your place in the queue to be verified, and the delay - even if you add schools after you are verified - is negligible. In the meantime, research the essay prompts in the School-Specific forums here, and crank the essays out. once you're verified and the rest of your secondaries come in, you can turn them around the same day you get them - copy and paste, baby.
if you do this, there's no reason you can't be complete everywhere by the end of July, maybe early August for the schools that are pokey about sending secondaries out (I'm looking at you, Dartmouth)... and that's not late.