If you don't have a heavy courseload your last semester (only taking physics 2, for example), it's possible to study for the MCAT at the same time and take the MCAT a few weeks after your finals. But with the emphasis on getting your app in on June 1st, those last few weeks in May are better spent working on the primary.
If you have the time to wait, I'd study that whole summer after finishing school and take the MCAT in August or January if you need to study for six months. Apply the following June. I tried to do it all last summer and it was too much for me with everything else I had going on. However, lots of people do it successfully. You would be best off taking the MCAT in early May, working on your primary the rest of May and getting it in as close as possible to June 1st, and you would theoretically have your MCAT score and be verified before the end of June. Taking the MCAT in early June pushes everything back but that would mean studying for the MCAT and working on your primary at the same time.