There is a thruway bus from New Orleans and Jackson to Mobile, but no connections are available to the Crescent. If you are coming from somewhere in the Northeast, you could go via Chicago and the CONO to connect to the bus.
The buses are operated by Greyhound, but not all Greyhound buses are shown on the Amtrak website. Going southbound, the Crescent actually arrives 2 1/2 hours before the last bus to Mobile leaves, but that is apparently too close for Amtrak to guarantee. You could try that but would need a backup plan in case the bus is missed (likely either a rental car or hotel and bus the next morning).
Going north, there is a relatively safe connection from Greyhound to Amtrak at Tuscaloosa, but they stop at different locations and the tickets would need to be booked separately.
driving times depend on I-10 congestion. Took this poster one time 4 hours from I-10 / I-12 intersection to downtown. Often better to take I-12 to the causeway even though longer. BTW to avoid New Orleans I-12 much shorter.