I had to erase and reinstall macOS on my MacBook Air while in recovery mode. I was able to reinstall but I’getting hung up in Migration Assistant. My Time Machine external drive is not being found to begin the transfer. Anyone have any solutions?.

I have had this happen on the last three erase and reinstall with different Macs, whether connecting by Wi-Fi, USB, or Ethernet to a network drive or to a clone. What worked for me was to ignore the possibility of using Migration Assistant during the reinstall, complete and update if necessary, then ensure that connections are secure and the drives are visible and accessible before running Migration Assistant from the Utilities folder.

How exactly are you trying to access the drive? If over Wi-Fi, that may be the problem. Wi-Fi is fraught with problems.

Of course you can shut down and restart, in fact that's likely necessary. Is your drive available on your network? That's always worked consistently for me.