As someone on the other side, in IT support, you can fix this yourself and I wish more people would.
Before your old phone gets wiped and sent to the graveyard, log in using authenticator, and go to "view account" from any of the online pages for Microsoft (if you're unsure, try login.microsoft.com ). Go to your security options, and you should see all the info you need to remove the old authenticator and add a new one.
From here you can also add backups, which I encourage everyone to do.
It saves you from having to call IT all the time to fix it, and since you don't have to go through the usual back and forth of verifying who you are, or whatever, and getting them to do a thing, you can take care of it for yourself, by yourself, without those unnecessary delays.
Your IT people will appreciate it, and you'll have to talk to them a bit less as a result.