What version of TS and is your licensing user-based or core-based?
Am using TS version 10.5 and user based licensing. Let me know for any inputs.
It might be worth a quick shot to bring up the "Manage Product Keys" app and then highlight the existing key and try the "Refresh" button to see if it updates the Maintenance Expires end date.
If if doesn't, then you could "de-activate" the current key and then "activate" the new one:
It really shouldn't be necessary to have to do an uninstall/reinstall just to update the license key.