A month? Really? You may not have seen the actual AMCAS, but trust me, you can fill out everything but EC's and PS in one sitting.
I spent maybe 3 hrs. yesterday filling out all my course work, picking all my schools, writing biographic information, etc. I made sure everything was accurate and then started on my ECs. Now EC's will take some time, because, at least for me, I am trying to come up with a 2-3 line description MAX of my work. I could write an anecdote but this is a bad idea, since it forces readers to spend a lot of time reading worthless info (unless 1 or 2 EC's are EXTREMELY UNIQUE, avoid long descriptions, and even then keep some stuff for interviews)
Now your PS should be in the drafting stage at least. If not, take an hour or 2 every day to write, rewrite, edit, rewrite, scrap whole concept, write and so forth...Honestly though, other than the PS and EC, you can finish AMCAS in a couple days. Try to do this before June 1, and if you feel like PS is strong, submit on the first day. This is my plan, and I'm taking the June 18 MCAT. I am going to submit the very first day, and then spend the next 2.5 weeks on final prep for MCAT. Once MCAT is over, I'll wait for verification and secondaries and start on those as soon as possible. The problem with waiting til after June 18 to start your app. is the PS SHOULD take a long time. You have 5300 characters to make yourself memorable, do you really want to rush that?
You'll be fine taking an hour off a day to do application stuff for the next month. Just fill out everything, have a friend take a look at it and make sure your info is accurate (pick someone not taking the MCAT or preferably not in pre-med...my business friends seems to have a lot more free time). Come June 18 do really, really well, and prepare to start your secondaries. I think this way you'll be less stressed and an early applicant. Good luck!