Well basically the old garage is 2 concrete boxes and we upgraded one - walled up the passage between them, rebuilt walls where needed, new surface, new floor, the only opening is now the (steel) door whereas before the super wide super old wooden door and walls cracked in places gave massive opportunity for mice etc to get in. OK a badger would have been too large but you get the idea.
So now, yes in principle mice could still get in through the open door, but we keep it closed and we keep a set of traps on the inside, the very same ones you use, and baited in the very same way. From our experience with another room nearby this should do the trick, and if not, there's always poison, I'm not at all happy to use it but it really really works. (Ofc no other animal should even be able to come close.)