On a recent trip to Atlanta (southbound), when we arrived (on a Saturday) there were long delays in getting checked baggage off the train.
I opted to take a cab to the Enterprise immediately and wait there for them to open, rather than wrestle with my baggage into the cab and at the rental agency.
Good thing too, we were first in line for a car and when I walked the parking lot behind the building to see what we'd get, I noticed a distinct lack of vehicles. Turns out they hadn't gotten a single return the previous night (they were all extended) and an impromptu 'order of arrival' list was written, with my name at the top. An hour after opening, someone finally dropped off a car and since we'd rented to that class, it was ours. We drove back to the station, collected our checked baggage from an empty (except for the attendants) station and were on our way.
When we returned the vehicle a few days later, it was the same attendant, and he said he finally got cars for everyone who was waiting just before they closed - if we hadn't bailed we would have been waiting quite a while and blown the day in an Enterprise lobby.
Amtrak lets you pick up your luggage up to 24 hours after it arrives with no charge. Take advantage of that!