Here are somethings you can do:
1) Go to career websites for every research institution near you and submit apps. In big cities there are many positions, but many applicants too, so carpet-bomb them with applications. Personalized apps to jobs you qualify for should get you a few interviews.
Edit: Yeah you will get a very low response rate. I submitted over a 100 apps to a dozen different medical schools and only got 2 interviews.
2) Go to institutions you are familiar with and look up researcher profiles/websites. Sometimes they advertise open positions, so directly email them with cover letters and CVs.
3) Contact your old PI and see if they have a paid position, or if they know if any of their colleagues do.
4) It might be too late now, but have you considered the NIH IRTA? Applications are taken all year round. Most new postbacs start June-August., so you should submit an app pronto.
For all of these things, you should start doing 6 months ahead of anticipated start date. So if you submit now, except to start in August-September if you get a job offer.