Planning a road trip in the southeastern USA for next fall. Wondering if there is any "best way" to rent a car for three weeks with pick-up and drop-off in different states. Thanks.

When we wanted to pickup a car in Memphis and return it to Nashville (or vice versa ), it cost $1000 more than picking up and returning in the same city.

Pick a loop and do the cities/areas in order. What city did you want to start with to fly into? This is assuming that you are a US resident and not someone from outside the US that gets to do one way rentals without a drop fee.
How many in your group?

Unfortunately as US citizens, we don't get the free one-way rentals without a drop fee. I really don't want to do a loop - we want to fly into New Orleans and travel to Washington DC (hitting Savannah, Charleston, and then the Blue Ridge Parkway) and then fly home (to California) - making this trip a loop would add too many extra days.

Look at Enterprise.

I have had good experiences with their flexibility.

Anything you want to see in Alabama or Mississippi?
Do a 4 day car rental loop from New Orleans after seeing the city walking and riding the streetcars.
After seeing the things outside New Orleans turn in the car and ride the Amtrak Crescent to Atlanta (leaves NOL at 7AM).
See Atlanta for 2 days and then rent a car for the SE US for about 10 days getting as far north as Washington but returning the car to Atlanta and flying home from there.
If you wanted to go farther on the Crescent, you could go as far as Richmond and rent from there to see Washington and the BRP and some of the Virginia coast.
If Enterprise will work with you on a no drop fee car, make sure they don't just charge you a high rental to start with.

"Unfortunately as US citizens, we don't get the free one-way rentals without a drop fee."

Nonsense. There is no such thing as a citizenship check when renting a car. What am I going to show them, my passport?

I frequently use AutoEurope to rent cars one way here at home in the US. Try it.

Good idea, J62 - I will try that.

The extra charge is not, however, identified as a drop-off fee; rather the price is about $1000 more than if the car is returned where the initial pick-up took place.

I did try Enterprise at sites other than airports - found the total price to be way less than the big rental agencies charge.

Thanks for the suggestions!

When I rent a car in the US, I have to show my drivers license which has my picture and my address on it.
The companies that rent to foreigners rent the cars with the insurances including liability.
My most recent car rental was with Enterprise (7 days for over 3300 miles). Where I rented the car was about 100 yards from an Amtrak station (Galesburg IL).
I always carry my proof of insurance card to show when they try to sell me insurance at the rental counter.
If there is a gas station within 2 miles of the car rental place, do not get suckered into a gas plan.

My only tip is to price monthly rates (if available) when shopping for a rental car. One month at a monthly rate can be substantially less expensive than three weeks at a weekly rate.

If you have the flexibility, check to see if there is a difference in cost if you do the trip in the opposite direction. Sometimes the companies want cars delivered to a particular location and will not have the one way fee.