For re-release/re-dispatch purposes... at the time of that re-dispatch, the alternate minima required should be that which would have been used originally, at the initial flight planning stage, prior to departure.
If no alternate exists which meet that requirement, the flight should divert to the non-redispatch destination.
The above is how it works at all carriers where I have worked, that use re-dispatch/re-release procedures.
Now, I know one Captain, that did not follow the above, then found his destination, and nearby alternates below minima, so had to divert to a very distant alternate, and after landing with his B747SP, on the taxiway, 2 of 4 engines flamed out due to fuel exhaustion.
The 747 fleet manager was
not pleased...especially as the concerned 747 Captain was the director of safety for the airline.
