I've been helping startups as VP Product and am now considering freelancing. Whether you're early stage or growth stage, I can see the need for a dedicated PM, but a good PM in whatever shape or form is way better than a dedicated average PM. So I'm curious - would you hire a freelance PM to help you out ?
I know of companies that hire contract Product Managers. I've interviewed a number of PMs over the years and otherwise been involved in PM hiring decisions while leading engineering and scientific teams.
Although most firm prefer W-2 (direct) PM hires for a number of reasons (including legal), sometimes the applicant will request 1099 status. They might be former W2 employees we are now part-time in business school or raising a family, and can't commit to the full-time workload. There might be temporary Visa issues that allow 1099 but delay a W2 hire.
The most common reason I can see is if the firm is a start-up that is rapidly scaling up and doesn't want to commit to a full-time hire until either they've secured additional funding and the person has proven themselves in the lighter 1099 role (or possible both). This is the way some temp hiring works: the employee will be on a W2 from the temp agency, but is a contractor to the employer. That's a route some better funded start-ups will pursue if they're desperate to make a PM hire, but don't want to take the legal risk on a direct hire. Later, when finances are more certain, they might convert the temp to a direct hire.
It doesn't sound like you're in any of the situations I just mentioned. The term "PM" potentially encompasses a wide range of skill sets. It could be anything from someone with some simple time-management and prioritization skills all the way through very technical or client-facing PM roles that may require valuable technical, sales, marketing, branding and management skills. You mention you're former "VP Product" so it sounds like you're probably at the high-end of the PM skill range. Therefore, it's doubtful you could find a position through a temp agency.
Start-ups that are rapidly scaling will try outsource their non-core functions, including more senior-level positions. You have firms specializing in outsourcing entire HR departments, business development departments, engineering and IT departments. See advertisements for start-up advisors on places like Clarity that say they are "a timeshare CTO."
So those are some strategies you could try. You could work as a project manager for an outsource engineering or IT firm that presumably requires an elastic workforce to handle client requests. You could try rebranding yourself as an "timeshare Chief Product Officer" that saves start-ups money by providing them with a part-time experience product person at a fraction of the full-time hire rate.
I have experience in a number of areas, and have started on Clarity by listing myself as an expert in analytics and predictive modeling. But I've interviewed hundreds of engineering and PM candidates over the years for extremely selective employers, and have worked closely with PM roles while leading my teams. I'd more than happy to take a call and answer any follow-up questions you might have.