I spoke on the phone with Amtrak's director of Interline services.
He thinks that Thruway connections to and from Birmingham will be a low-hanging fruit when the new station opens.
A Montgomery connection will be in place right away because the existing Greyhound schedule coincides perfectly with the train schedule. There's just enough lead time in both directions to make the connection reliable.
A connection to and from Nashville can be established on day one with the Crescent going to and from Atlanta and the Northeast. Unfortunately, that connection would be nowhere near as appealing as a connection with the Crescent going to and from New Orleans. In order for a Greyhound/Amtrak interface to work on that city pair, Greyhound would need to run an earlier southbound schedule.
A Birmingham-Memphis connection could reasonably connect Birmingham with the City of New Orleans and connect Memphis to the Crescent. It would also connect Tupelo, Mississippi to both routes. However, Greyhound would need to run another round trip schedule and make an additional stop at Memphis Central Station.
I hope that the officials at Amtrak and Greyhound can work out the details to take advantage of the facility.