Without an MCAT score, it's premature to develop a school list. Keep in mind that "Christian" schools, like Loma Linda, and the Jesuit institutions: Creighton, SLU, Loyola, and Georgetown, have a mission to train docs with humanistic ideals. You haven't specifically mentioned any long-term nonmedical community service to those in need (which may or may not be blended in with the 600 volunteering hours), which they'd be looking for (without regard to your religious affiliation, in the latter instances). Texas schools find that type of experience appealing as well, BTW. As would UUtah, and Rush, and USCalifornia.
The first week of June is the soonest you can submit the AMCAS application; it is not the deadline date. You can fill it out starting early in May. You will have your May MCAT score back before any school receives verified applications from AMCAS (traditionally at the end of June), and they will tell you the percentile rank where your score lies a few weeks before that, so you can target suitable schools before the score is known. If you submit early, to, say, one school so the transcripts are verified, you will have zero disadvantage.
Do you have some dedicated physician shadowing?
Which Hispanic community do you represent, and what is your level of Spanish-language fluency?