Best recruiting approach for my non-traditional background?

To get to the point, I have a non-traditional background. Transfer student, terrible GPA (worked full-time through school), and have taken non-traditional careers jumps... for example, I only spent a few months in IB and hopped to PE and have been in PE now for almost two years... I haven't had any success with head hunters... not while in banking and not now... I have always landed jobs through unique situations... cold outreach, etc. I've had an interview or 2 at L/S funds through random linkedin postings but haven't converted due to my lack of experience.

What would be the best approach to break into a long/short HF in my current position? If I knew the types of memos HF professionals created I would plan to create my own and have a pitch and try cold outreach... but I feel disadvantaged in this regard due to my lack of experience. Should I try to hunt down someone who can provide an example and go from there? Curious to hearing other stories... my network is limited to banking and private investing.

And lastly, I've had an interest in the public markets since a young age and it's always been the goal of my career aspirations... I think my story will make sense to most, just need the opportunity to share it.