Using Classic Outlook on Windows for personal email, calendar, and contact management
It seems that the "New Teams" is a Microsoft Store app and it does not create a folder in %localappdata%\microsoft, probably because the "New Outlook" is a browser version of Outlook that does not need an addin for Teams integration.
I used the Bulk installer from here - https://learn.microsoft.com/en-us/microsoftteams/msi-deployment#msi-files
Please check your Office version (32 or 64 bit) before downloading or you might download the wrong installer!
I also had to delete the registry key also mentioned in the same article - https://learn.microsoft.com/en-us/microsoftteams/msi-deployment#clean-up-and-redeployment-procedure
If the installer does not start on its own, start it from Program Files\teams installer or Program Files (x86)