Approaching PE recruiting from a lesser known bank?

I'm an incoming IBD analyst starting at a bank that people on this forum don't really talk about- not an EB/BB/top MM, or even one of those specialty banks that get a lot of credit on here. To give a rough comparison, I'd compare it to something like AGC partners in that its pretty regional, not in NYC, and not doing huge deals, but does actually have deal flow. 

I have a semi-compelling story of how I ended up at this bank, but the truth is also that I simply didn't get ready for recruiting early enough and struck out at more mainstream Wall Street banks. 

However, I'm very interested in MFPE (and I'm not just some college kid that thinks that's where you make the most money- I've done my research and I believe it's the best place to start my career in investing for a number of reasons). 

I understand that coming from a smaller bank, this is an uphill battle. Looking at past analyst exits from my bank, I see people going to a lot of no name PE and growth equity funds with small AUMs. That being said, I think I have a bit more of a competitive resume. I go to an ivy, I have a perfect score on my SATs and a 780 on my GMAT, and I think I have a compelling story as to why I'm interested in PE, specifically at a MF. 

Is there any way to best position myself for PE recruiting to a larger fund? I've been working on my modeling and am very confident I can show my modeling skills are up to par with someone from a BB. My biggest fear is simply not getting looks from headhunters and missing the boat on recruiting- can I be proactive and reach out? If they talk to me and are impressed (and I can show that I'm 'better' than my bank), is it possible to get put in front of a fund? I have complete confidence that, if I'm put in front of a fund, I can show that I am a qualified candidate, but my biggest worry is that I won't even get interview looks because of my background.