this remains a blackbox even for people who get faculty positions. Basically, what happens in that if you get a Cell paper, say, or work for a really famous person, you'll get noticed by some dept chair because there aren't that many people who work in your field and does these things. You get hired automatically. (least that's what I heard.)
If you apply for a job posted on NatureJobs, say, then you are facing like literally 1000 other candidates.
You don't need to do a post-doc after fellowship to get hired as a faculty. The catch though, is that you don't have "tenure" necessarily.
I find this whole idea of doing a fellowship and a residency and THEN a postdoc absloutely ridiculous. that'd be 10 more years after MD PHD!