In certain situations I respectively disagree with the previous poster. IMHO, a last minute letter (or even better, a phone call) to your program of choice might help your application if:
the letter writer/caller actually knows you; thus, this hsouldn't be some alum you've ferreted out just for the occasion but someone who can really attest to your potential
thus, in reference to the last statement above, the person should be a medical alum from the school. If its simply someone who graduated from the school but in another discipline I doubt it will make a significant difference in your application
Of course, if the above conditions were met, I might have assumed that you would already possess a letter from that person. If this is so and they have something new to ad to their previous letter, then perhaps an update might be in order, althouhg I tend to favor the chatty phone call (this is said to work especially well if the parties involved know each other and the caller can tell the callee that "Joe has just returned from his interview at your program and was quite impressed...blah blah blah".
If you can't meet the above conditions (letter writer who knows you well and is a medical alum of the school) I would say that a letter is not likely to help your application any at this point.
Best of luck.