PE to HF

Currently a 2nd year Analyst in a coverage group at a BB. I'm going to be joining a relatively new / start-up PE fund with a specific vertical focus. I understand PE becomes much more relationship management, execution focused, etc. over time but wanted to get PE Associate experience. Given I'm not at a MF, how often do you see people move to HF / public equity space (assuming they are competent, intellectually curious about investing, want to focus on I-Thesis vs. closing deals, investing stans, etc.) and what are the commons routes? Do you go through headhunters again? I know MBA might be an option but would like to avoid that if possible and haven't seen too many go from MM PE to HF.

Also are mutual funds and L/O an option post PE? Understand this is a broad question but want to understand broader landscape first before I narrow down on a game plan

Thanks!