Also to clarify things, the "undiscovered" area(ie fogged over on the map) needs to be a good distance from your uncovered area already. If you, for example, uncovered the area around a mountain but not the mountain itself, there is a high chance that no caves will be there even if on a new version of the same map that there normally would be.

The system is more liberal with what it considers rendered previously for a map than the actual fogged out area.

So as long as the mountain biomes are decently far from exposed map areas, and they're meant to have caves(as not all mountains do) then there should be no need to hit a new map.