I don't believe there are any car rental locations near there. However a $15 or less cab fair will get you to the Lynchburg Airport which has rental cars available during the times that trains service the Lynchburg station. All the times I've been at the Amtrak Station there have been cabs nearby. From the station the drive to the airport isn't but perhaps 7 miles.
enterprise
There are several rental car companies near the Rocky Mount Amtrak Station. The major rental companies are all within a few miles of the train station and approximately a 10 minute or less drive.
The closest car rental company near Kingston, RI Amtrak station is Enterprise Rent A Car located at 120 Point Judith Rd Unit 3 in Narragansett, RI. There aren't any rental car companies in Kingston, RI.
Lynchburg, VA. From Lynchburg, you can catch Northeast Corridor trains to Washington DC, Philadelphia, New York or Boston. You can also catch the Crescent south to New Orleans.
There are no actual car hire (rental) offices at Amtrak's Lynchburg, Virginia station. However, Hertz, Avis, and Budget are happy to accommodate passengers. Bookings are to be made for the city of Lynchburg, noting that the passenger is arriving via Amtrak. Once the passenger arrives, they phone the rental office and either of two things happens: They will send a driver to collect the passenger, or they will instruct the passenger to take a taxi to their downtown office, obtaining a receipt for the cost of the fare and the cost of the taxi will be deducted from the final contract settlement. There still remain a few destinations in the US, where Car Hire companies will hand-deliver the vehicle to the destination and await the passenger's arrival. However, this has now generally been limited to the car hire company's 'Elite' membership frequent renters cards, such as Hertz Black, Avis Wizard Gold, etc.
The closest Amtrak Station to Montgomery, AL would be Birmingham, AL which is served by the Crescent. It is 1 hour and 28 minutes away or 90 miles.
Dwight - Amtrak station - was created in 1891.
Norman - Amtrak station - was created in 1909.
Kissimmee - Amtrak station - was created in 1883.
Alliance - Amtrak station - was created in 1990.
Hanford - Amtrak station - was created in 1897.
Gilman - Amtrak station - was created in 1986.