Post-doc applications can be highly variable in terms of timeline, number of applicants, and firm deadlines (as you noted). If it's a top lab in a desirable location with an advertised, funded position, there are likely to be a large number of applications. For less well-known/younger PIs they may only get a few. In my (limited) experience, I can't think of anyone who DIDN'T get a post-doc that wanted one. With enough looking, I think they can be found. The other option is to just send an inquiry email to someone you are interested in working with. There may be internal opportunities for funding, or you could apply for an F32 (or other mechanism) to fund yourself. These positions aren't advertised but I do know of people who have gotten positions this way. Finally, as you've probably noticed, the posting of ads for these positions varies quite a bit. You can likely expect a batch once the next round of NIH funding decisions comes through. I can't say when you'd expect to hear back though — likely largely dependent upon the number of applicants etc.