Advice re: PE recruiting / headhunters

A bit about myself. I had a career outside of finance for a couple of years. I dealt with PE firms along the way and really liked what they do (namely, the business decision-making and being in charge/managing of their investments, as opposed to servicing and advising clients). I made a decision to do my best to get into PE, and I have recently lateralled into a pretty good IB group (pre-MBA) where I think there is not-too-bad a shot at PE. However, given that I am a lateral hire at the IB group, I don't think I will be on the headhunters' radar unless I reach out to them myself. Once I get in touch, I think I have pretty good answers for why I want to go to PE and my story, which is what I assume they would ask about - but I need some advice on how to get in front of headhunters so that when the time comes for December/January PE recruiting season, I have interviews lined up.

I know of a number of the headhunters usually mentioned - CPI, Dynamics Search Partners, SG Partners, Henkel, Amity, Oxbridge, Glocap, GoBuySide, (maybe SearchOne Advisors, Ratio Advisors, Bellcast Partners). Please let me know if I missed any other good ones for PE. How would you recommend I get my name in front of them at this stage? Should I email their firms' general email addresses letting them know who I am and that I am interested in speaking to them? Should I pick out one person in particular and email him/her? Should I cold-call him/her? If so, how should I determine which person to pick out? Am I thinking about this the right way?

Happy to give more context in a pm. Appreciate the help.