Are there ways to to gain a meaningful edge / backchannel into a top BSchool?

As a very generic (Target Undergrad --> IB --> PE) MBA applicant whose odds of getting into HBS/GSB/Wharton are probably a coin-flip at best, I'm wondering what it takes to gain an edge in the application process through off-resume/off-application actions. Specifically thinking about any anecdotes where a Head of the Firm / Partner-type is willing to "make a call" for an associate.

Who would the person have to be to make that a meaningful edge, how would you go about asking said person to do that for you, is that even a viable course of action to consider?

I'm sure this is a very nuanced topic that could depend on many factors including alumni status of the person (not an alum / undergrad alum / MBA alum), fund relations (whether or not the school is an LP and the size of their commitment), etc. but I'm really curious to here if anyone has a story they can share about this.