Well, a lot of people here complete their secondaries in August and early September and have a lot of success. I got plenty of interviews (11) by completing mostly in the last week of August and first week of September. And I mean COMPLETING (fill out the secondary form, send them a photo of yourself, send them your letters of recommendation, send in anything else they want ($$$), and then wait a couple of weeks until they sort through the massive amount of crud they are getting from hundreds of anxious overachievers and perfectionists who call 3 times a day to see if they are complete yet). This means having your recommenders write their letters in the early summer (some of them will go out of town and need more time/prodding).
My suggestions after having gone through the process myself just recently:
As soon as you can (early June), open up an AMCAS account and send in your transcripts from all schools you've attended and request letters of recommendation from all parties involved (you'll have to send your LOR to AMCAS, and it takes them a long time to process them, even if they are sent electronically. you might even want to ask them at the end of the semester, before they leave for vacation.) If you feel like you've had enough time to study, I would suggest taking the exam in mid-June like you planned (earlier would be better, but not needed), and that way you will have your score in mid-July (current turn around time is 30 days). While you are waiting for your score, fill out your AMCAS application and decide which schools to apply to. Once you have your MCAT score in July, you can submit your AMCAS application, and then it should take them about 3 weeks to verify it (the longer you wait to submit, the longer the line and the longer it will take to verify, so don't delay). While you wait, make sure you have all LOR, and send them in as soon as you get them (I would suggest using something like Interfolio to manage your LOR, I found it very helpful and easy to use). While you are waiting to be verified, you can start writing any essays you will need for secondaries. That way you will start getting your secondaries to fill out in August, and can have them done by the end of the month. This will put you ahead of the game, since schools will sort through your stuff by mid September or so, without being SUPER early. Although being super early isn't a bad thing, I'm not very convinced that it has any benefits. By being complete in September, you'll be far ahead of most others, and you'll be among the first to be reviewed.
But be warned: the month that you are waiting for your MCAT score seems like a long time to fill out the AMCAS, but it's quite a pain in the butt. Don't wait until the last couple of days thinking it's like any other form. You have to put in EVERY CLASS YOU'VE EVER TAKEN by hand (which was a pain for someone like me with over 200 credits to my name before senior year). You also have to put in all activities and experiences and family info and life info and AHHH!!! (I would actually print out a copy of the AMCAS and keep it handy when filling out secondaries. I don't know WHY but many secondaries basically just ask you to put in the exact same info over again. I found JHU to be the most annoying culprit in this way, but they are JHU, so they can do whatever the crap they want.) Also, put a lot of time and effort into your 2 big essays. It will really help you a lot at some schools to have a stand-out essay. Write about what makes you unique. The key is to make them want to meet you. If you are just like everyone else, they will not want to meet you (aka no interview), but if you are different, that is interesting (but don't be "interesting" by being weird or creepy.) I wrote about an extra-curricular activity that I do, that no one else has every written about, ever in the history of MD/PhD programs (as I have been assured by many a program director), and it's nothing very special if you ask me. Find what makes you unique and describe it as it pertains to how it will make you a better physician and scientist. Don't just rehash the rest of your application.
Good luck!
(hmm.... this seems like a useful guide to put up somewhere, since I didn't know any of this stuff coming into it. I just got lucky with my timing, or I would never have known to sumbit early. maybe I should clean this up and put it somewhere where others can see it)