Aside from the Bridge(and it's possible there was a set piece in the Bridge ensemble that qualifies because it appeared outside of the Bridge) set pieces, which Star Trek TOS structural interior set piece garnered the most onscreen time? For instance, was there a specific door or wall panel, or corridor or ? that had substainaly more air time as the crew scurried about the Enterprise than any of the other Enterprise set piece interior props? Which set piece was it?
It would be difficult to say with any confidence without going through and timing each scene, but having recently rewatched season 1, my guess would be the set for crew quarters (usually Kirk's), which often featured in extended dialogue scenes in many episodes. If you google "quarters" along with site:www.chakoteya.net/StarTrek you can find a large number of episodes whose transcripts mention that some scene is set in someone's quarters...some are animated series episodes, but you can add -animated to the search, and the search results show 44 transcripts.
For "turbolift" there are 31 results, but looking over some of the examples and doing control-F (or command-F on a Mac) to search for where in the transcript the word appears, the dialogue scenes set on them usually appear pretty short, while the ones in quarters are often longer. For "transporter room" you get 59 results, but searching those transcripts there seem to be a lot that just feature Kirk on the bridge giving an order to the transporter room (only if the transcript features [transporter room] in brackets is there an actual scene there), and when there are scenes they again seem to usually be pretty brief. For "sickbay" there are 44 results, though again there are a good number where sickbay is just mentioned in dialogue rather than a scene being set there...when there are scenes there my impression is that they're likely to be longer than typical scenes set in the transporter room or turbolift though, so my guess is this would be the second most likely candidate for the non-bridge set having the most airtime.