But you'll probably be limited in your options. Most schools have begun accepting students on a rolling basis, and thus the total "available" slots are shrinking.
But I know for a fact that our school was still conducting interviews late into June (and several of those folks ended up being my classmates), and I hear that students at other schools have interviewed as late as July.
If you're pretty flexible as to which program to attend (and just my opinion, I think virtually all of the osteopathic schools are fine institutions to train at)... then go ahead and see what happens. Each year you put off from your education is one year you're taking away from your future salary! I recommend calling their admissions office and maybe shmooze the secretary into giving you the heads-up as to whether or not you have a legitimate shot at an interview/acceptance.
Something else to consider. Because of the rolling admissions... you're also going to have students that were accepted early, paid the deposit and still pulled out to attend a different school. Happens all the time.