Caves an mines can be good shelter however one must have the means to test said mine, seeing many gasses still can be present. Radon, amongst other things, is a serious concern and not easily removed or shielded from. At least a Geiger counter to read radiation both inside and out in the event of a nuclear event, And like amateur radio it takes learning and practice to know how to properly use the equipment. In bad weather some mines fill with water spreading contamination as well, I had a friend living in one that the entrance had a permanent pond to pass through to get in. he could not have any fire or candles going because the smoke had no where to go. Problem to with mines and caves is the possibility of earth quake bringing it down on you. Grant you many have already stood up to several quakes, so in theory they've settled already. I lived at Big Bear Lake California several years and through several quakes as well, and several hikes through mines and caves and natural boulder caves too. Another thought is the fact that these mines are not secret, and though you may not have seen any one while your there does not mean others don't' have the same experience at the same place thinking the same things. If the road was not made by you, chances are others know about it as well.
If you're young and healthy raydon appears to probably be bad. If you're have problems it may not be all bad. The most fascinating think about radon therapy is an actual clinical trial has never been preformed.
Grand Saline salt mine isn't too far. Access is by a little elevator that goes 50 stories down. That sounds like a one way trip if topside infrastructure is lost.