I would say no they can not use their second action to mine from the adjacent mining station.
From the rule book under phase 5:
"Captains are resolved in the same way as Crewmembers, but you may activate the effect of their current Station up to two times instead of once. You do not have to resolve you Captain first, but when you do resolve your Captain you must decide whether to activate the Station’s effect once, twice, or not at all; and you must return the Captain to your Ship before resolving any remaining Crewmembers."
And then under rover station:
Rover Station – One of your Crewmembers assigned to a Mining Station may collect resources as if they were assigned to an adjacent Mining Station.
I read that as that you have to choose to active the Captain for the station that he is assigned either one, twice or not at all. If you choosing to activate it once you can not then go back and change that. So with using the rover it is the same, but the captain is acting as if they were at the adjacent station, and not the one that they are currently assigned.
Hope that helps.