Yes, that was my initial inquiry. Apart from uninstalling and reinstalling, is there any other method available considering I'm already logged in at this point?
I've seen some other suggestions that appear more like ripping the data out from under the Authenticator like clearing the app's data and cache, but based on this Microsoft Authenticator Q&A, the best method is likely to click the Microsoft account on the main Authenticator screen, then click the 'Settings' gear in the upper right-hand corner, then click the Remove account button displayed there.
Common questions about the Microsoft Authenticator app - Microsoft Support
What I'm uncertain about and couldn't find a clear answer for, is whether the previously setup Microsoft account might cause any problems on the server end of the connection. Assuming that recovering the previous Authenticator data to the phone includes the private key along with all of the individual Microsoft and other accounts originally setup, that should reset the phone end just fine, but I'm not sure how the server end of this relates to that, though if it just uses your Authenticator's public key, then I guess that should work fine.
The other thing I'm unsure of is how you'll know which Authenticator backup to restore, which if you hadn't set that back up using the new phone yet, should obviously be from the old phone. If both backups exist, I don't know if it will offer them by name or only offer the new one that matches, so pay attention to that to ensure you choose the old one that contains the accounts.
I'd also make sure you have multiple other backup verification options set like phone number (text), alternate email address and even recovery code to ensure you don't get locked out if something goes wrong.
I did this process the 'right' way when I replaced my phone recently, since I always read before I do, but I understand how you can end up in this situation, so I'm surprised there isn't more about this issue that's easy to find, but I sure couldn't.
Rob