Ok here is my idea for quickly generating large underground volumes extending through three dimensions.
Divide an a4 sheet into quadrants.
Quadrants are either top-down or vertical.
If vertical then top is always at the top. If horizontal then North is at the top, the other compass points arranged as usual.
If you want, one vertical compass can have Up, Dwn, North, South instead, but that is a bid to mind-bending and crazy for right now so we will leave it out.
You need about three passage-forming elements. We will start with the simplest, a river. In blue.
Remember in the vertical sections a river can only go down, never up. Now we will add a fault.
Now a mine in green.
There are four kinds of hex, depending on how many lines are in the hex.
No lines - the wilderness of stone. Hard travel, have to search, go cave-by-cave.
One - Transport possible along route. Semi-wilderness, encounters rare but nearly unavoidable when they do happen.
Two - More travelled. Tribes. Small outposts. More encounters, can be avoided.
Three - Likely encounters, organised polities, armies, population centres.
So, depending on what hex you are in you just describe what you see on the map.
"A river leads straight down, a fault leads down to the west and up east, a minework leads up and west."
You can still search to go anywhere but those are the main routes and should (eventually) take you somewhere.
Encounter density probably equivalent to desert. Routes will compress travellers but population much lower overall so should even out.
Will do a table for the kinds of lines you can roll for tomorrow.