Can you demonstrate value to a potential PE fund instead of sending over resume?

I'm coming from a big 4 audit background in a non-major region for PE and VC firms. There are a handful in the area but no large ones. Many of these firms only have a few staff based on the company's websites, so there aren't really many job postings or openings. I'm wondering if I can demonstrate value that would entice a potential employer to essentially create the job.

So in thinking how to do that, I'm wondering what I could do to show that I would immediately add value. The only things that I can come up with are volunteering to cold call companies they may want to acquire, or to do research on companies in the area as potential targets and develop quality proposals. Would either of these ideas be even remotely useful to the firm, or at least make me stand out?

The problems with this would be that I would have to cold call during work hours (lunch break?), which is pretty risky. Second I don't really know how I would develop a proposal on a private company, but I guess comparative analysis with a similar public company?

Edit: Thinking again, no firm is going to allow me to cold call on the PE/VC firm's behalf, so that is out.