If you count ENT's "In a Mirror, Darkly, Part II", the slanted tube leads into a rather shockingly spacious network of full-size service corridors. Personally, I never accepted this idea of "secret tunnels and trap doors". It looked way too over-the-top, part-techno-geeky, part cloak-and-dagger.
I always assumed that the tube was just what we saw of it. It's just an access crawlway, built along the contours of the ship's infrastructure, to access machinery (or conduits, or fuses, or whatever) embedded in that particular wall. I'm fond of thinking (no foundation for it) that the tube is located along one of the curves in the hull (like maybe the saucer undercut) where the hull becomes too thin for a full-height deck. That's where machinery is stored: in those undercuts. There may be sensors or power conduits or control computers and control lines (or whatever) embedded there, where they will not take away from living space. The tube would provide access to that equipment where a full-size deck isn't possible.
If you look at "Day of the Dove" and follow what Sulu is trying to do, it's obvious he is nowhere near the secondary hull or the warp pylons.
