I was going to try DeDe on our last trip since I had heard so many good things. I called and spoke to Deadman directly. Unfortunately, he told me that he is not doing single day rentals any longer. So, this might still work for you if you are arriving by ferry, staying a week and then flying out of EIS. If, on the other hand, you want to do as Stoney suggests and rent a car for a day to get from the airport or ferry dock to your hotel, do some provisioning in the morning and drop off the car the next day, then that may not work for you. As great as Deadman and his people are, I can't imagine it is very profitable for them to drop off a car at one end of the island one day only to pick it up at the other end the next, so hard to blame them. But, that is where National has been great for us, especially when flying in to EIS. We picked up at the airport, did our thing, dropped off near our charter base after dropping everything off at the boat and they gave me a ride to the base. When sailing with TMM, they even let us just leave the keys with the TMM office when we left to go sailing.
If you are arriving by ferry into Road Town, the Road Town office is at Maria's by the Sea, so a very short walk from the ferry dock. Renting with them has been as convenient as I can imagine is possible.
One tip: do pay attention and make sure any damage to the car is noted properly when you pick it up. We have never had them try to bill us for damage to the vehicle yet, but they do look it over more carefully than you might expect, being on ilon time and all...