Yes, unless you apply to totally different schools that won't remember you from last season. You shouldn't just assume that the only thing holding you back was being late, that's a disaster waiting to happen.
When you re-apply, they will want to know how you improved as a candidate for medical school, and you can't say, 'my improvement is that I turned in my application earlier' (you obviously won't say that, but you know what I mean). Get new letters, show how you improved in your PS, how you matured, how you learned, etc.
Also, don't be too confident to apply to lower-tier schools, the last thing you want is to be is left out in the cold, wishing you had an acceptance from the lowest ranking school.