PE Off-cycle Internships

Posted this in the IBD forum but figured I might as well post it here.

I'm currently a senior graduating this December (one semester early) from a top 10 school (not Dartmouth) and already have a FT job lined up at GS/MS IBD. I'd like to use my free semester next semester (from January to May) to explore a different area within finance since I've basically signed away the next two years of my life to investment banking. Ideally, I'd like to follow the PE/HF path after two years in banking.

Couple questions:

1) How do Dartmouth students usually find internships during the winter / off-cycle season (Jan-May)? Are they mostly attained through networking or does Dartmouth have a formal recruiting process for winter candidates?

2) What kinds of firms typically offer off-cycle internships? Would it be reasonable for me to reach out to mid-large private equity firms, hedge funds, or BBs? Places like Bridgewater, Silver Lake Debt, MS, BAML seem to have hired winter interns from Dartmouth but I'm not sure whether this would be possible for someone who doesn't go to Dartmouth.

3) Are these off-cycle internships only offered to Dartmouth students? Would I have a chance if I did some aggressive networking? Would they be reluctant to hire a senior who has already signed for a full-time job?

I'm just trying to figure out what I want to do with the time that I have next semester and would really appreciate any advice or suggestions you guys might have. Thanks so much in advance.