Back to the O.P.:
TOS did show us some interesting "miscellaneous work areas", some of which were obviously cobbled together out of other existing sets. Examples were the botany facility in "The Man Trap" (an obvious redress of Sickbay) and the ship's theatre/rec room in "The Conscience of the King" (another redress, this time of Engineering). Perhaps the most interesting cobble jobs were Auxiliary Control and the Weapons Control Room ("Balance of Terror"), both of which were apparently derived from Engineering as well.
The notion of exploring a ship's cargo hold is an obvious possibility. I would also have liked to see more science labs. Another thing we never saw in TOS was an airlock or docking port. But probably the most fun would have been seeing the fabled 22-man transporter room. (! ! !)
I would also have liked to see more crew work rooms, like a deep space communications room where specialists process data, decode messages and scan the spectrum to find unheard messages to feed up to Uhura's station.
It would also be great to see the under-hangar-deck facility, where shuttlecraft and probes would be stored, maintained, and assembled/dismantled.
With the exception of "The Tholian Web", we never got to see Enterprise personnel in spacesuits. If we could go back to revisit TOS, I would want to see Scotty walking on the Enterprise's hull, making repairs, in a space walk similar to FIRST CONTACT. I think it would be great to show the Enterprise out on the frontier, isolated from any backup, when the ship suffers damage in a daring mission. Then we could see Scotty trying to repair the ship on a tight, demanding schedule, placing himself and his team at risk to get the ship running again.
It would also be great to see a TOS-era version of the Travel Pod or Work Bee as well.
Another non-Enterprise aspect of TOS that was never properly explored would be Cyrano Jones' mysterious scoutship. What if Unit XY-75847, mentioned in "Errand of Mercy", had actually been Jones (or one of his peers) spying on the Klingon Fleet? There's a wealth of story possibilities there...