Let me make some assumptions here, please feel free to correct me.
You say you have 310 hours total: 100 of those were the past 3.5 hours at a 3.88.
(210 * 2.2) = 462 undergrad GPA quality points
(100 * 3.88) = 388 post-bacc quality points
Therefore, your overall GPA (assuming no repeats) is:
(462 + 388) / (310) = 2.74
Calculating your science GPA would be downright hellish, because I'm assuming you have a lot of science courses between an engineering undergrad and this 100 hour post-bacc. Now, a 2.74 is very low. I think you have a few things working for you, though.
1) Superior AND lengthy post-bacc record
2) URM (under-represented) status
3) Superior MCAT score
Now, some schools will screen you out with that sub-3.0 GPA, no doubt. I'm curious, though, if some schools don't have an option on their computerized "screen," that allows sub-par GPAs get through if coupled with certain other factors. Two things I could definitely see them letting a <3.0 getting by because of would be a very high MCAT (maybe > 35) or URM status.
Like others have said, lack of healthcare experience is the real killer to your app. It's pretty late now, anyway. You planned on applying now? Unfortunately, you'd be put on the end of the pile, since med school is kind of a first-come-first-serve thing.
I'd apply next spring, personally. And I'd apply to osteopathic and allopathic schools. Osteopathic schools could be a godsend for you. Osteo schools calculate GPAs differently than allo schools. When you retake a course, allo schools average the two grades (so an F, retaken with an A, gives you a C, numerically), whereas osteo schools only take the newest grade (so if you had an F, retook and got an A, AACOMAS says you got an A); it's a beautiful thing for those who made mistakes in the past!
So let's assume, since your GPA was so low (2.2 originally), you had a couple (3) F's, a few D's (3), and some C-'s (4). If you retook all those courses, and got A's in all of them (which you've proved to us you are capable of these days), you could potentially have the following cum GPA:
Total points - [sum of credit hours * grades] + [retaken course grades * credit hours] =
462 - 9(0) - 9(1.0) - 12(1.7) = 432.6
432.6 + 9(2.0) + 9(2.5) + 12(2.85) = 507.3
507.3 + 388 / 310 =
2.88 Allopathic GPA
Now, a 2.88 is still low and not really very helpful, right? Let's see what your osteopathic GPA would be, though:
462 -9(0) - 9(1.0) - 12(1.7) = 432.6
432.6 + 9(4.0) + 9(4.0) + 12(4.0) = 552.6
552.6 + 388 / 310 =
3.03 Osteopathic GPA
YES! We can get you up to a 3.0. I made a lot of assumptions, but I just want to show you it's completely possible, even with 310 credit hours, with the majority at a 2.2 average, to get your 2.7 up to a 3.0 in only 2 semesters!
So I say to you, retake as many F's/D's/C-s as you possibly can and get all A's. Hell, if you had, let's say, 10 F's, your allopathic GPA could actually be (after those 10 F's being retake with A's and essentially counted as C's) a 2.93, and your osteopathic could then be a 3.12.
Now, it's 1:06 A.M. here, so my math could be a bit fuzzy, but I'd say that next year, you'd have a VERY realistic chance at a DO school (>3.0 GPA entirely possible with retakes, far above average GPA, and URM status). If you'd like me to do more exact calculations and forecasting, PM me every class you've ever taken with the grades; I can do your overal, non-science, and science GPA's for you. I'm sure you'd rather not post them on a public forum
Good luck, man!