How would you map an enclosed corridor that is also a bridge between buildings?
In the past, I’ve just mapped them as a private footbridge, although with a level level as well as a layer. I’ve put the rest in a description. https://www.openstreetmap.org/way/299852432
Note that enclosed corridor bridges can exist at road level as well, not just as private connections between buildings.
I’ve not used it, but https://wiki.openstreetmap.org/wiki/Key:building:levels would appear to be what you need (see
diagram at the bottom of the page). How it renders is another matter. 
Another possibility for a simple geometry would be https://wiki.openstreetmap.org/wiki/Key:tunnel#tunnel.3Dbuilding_passage but
that only really works for what you’re describing with two-story buildings.
This is what I did: I drew an area and detailed it as follows:
highway=corridor
man-made=bridge
covered=yes
level=1
I would probably map it as a building part with min_level as Brian suggested.
Unless you’re doing indoor mapping for the corridors in the adjacent buildings, I see no reason to do it for the corridor running through the bridge.