The honest answer is that you often will not know in advance, and that is by design rather than by accident.
Prison transfers are not announced ahead of time to inmates or their families. The facility does not want anyone on the outside knowing when a transfer is happening because advance knowledge creates opportunities for interference, escape attempts, or security complications during transport. Your husband will likely be woken up early one morning, told to gather his property, and moved out before most people in the facility even know it is happening. He may not be able to call you until he arrives at the receiving facility.
The three months he has already spent in county jail are not lost time. Every single day he has been in custody counts as time served and will be credited toward his sentence once the official calculation is done at the diagnostic center. That credit accumulates whether or not anyone is keeping track of it from the outside.
The best way to track the transfer after the fact is through the state's department of corrections offender locator. Once he arrives at the diagnostic or reception center and is processed into the state system, his record will update in that database with his new location. Check it periodically starting now so you notice when the update happens.
In the meantime, keep sending letters. Mail forwarded within the same state system typically follows the inmate through the transfer, and letters sent to the county jail address will usually catch up to him at the diagnostic center even if there is a brief lag.
The transfer is coming. You will most likely find out when he calls you from wherever he lands.