How does PE recruiting work from MM banks?

Incoming FT analyst at a top MM shop (Baird/Blair/HL). There's a ton of old threads on here about PE recruiting from BB/EB or consulting companies, but can anyone shed some light on what the recruiting process looks like for an analyst from one of these respected MM places? Will headhunters reach out to me? I know its going to be tough, but I'd like to give MFs, or at least UMM PE, a shot during on cycle, and if I miss out then I might consider lateraling to a BB and trying again. I went to a target, have good GPA/test scores, etc, the only thing stopping me from being a competitive candidate is that I'm not at a BB.

Should I just focus on lateraling and trying to do on-cycle the next year? Would love any insight from someone who recruited for PE from a MM shop. From my understanding, a lot of the PE exits in the space tend to be referrals by senior bankers to their PE friends at sponsors they work with a lot, but from looking at previous tomb stones it seems like the typical exit from my firm is to smaller funds and I want to do something bigger.