You don't have to book a car in advance from the U.S. to get the best deal. Expedia offers you the option of using their U.S. site. While in England recently, I ran the identical search in two tabs -- one on the default UK site and one on the U.S. site. The prices on the U.S. site were about a third lower.
Do be aware that prices offered to UK residents are nearly always inclusive of CDW and those to USA residents are most often exclusive of CDW. The reason for that is they type of insurance that people routinely already have. There are exceptions on both sides, and sometimes currency issues.
We find AutoEurope based in Portland ,Maine to be the best resource for car rentals in Europe. And they stand behind you if there are any probems with your car or rental.
With the ease and speed of internet, i can price quickly compare almost everything, kayak, expedia, costco, direct with car rental companies, etc.
On my last trip to Hawaii, i got the best car rental using autoeurope. Go figure. I spoke with autoeurope representative and he told me autoeurope rents cars all over the world and had booked a large group of cars at a huge discount for promotion.
PS. As noted by OP, booking through different country website may make a difference. There was a difference in price between the US and Canadian autoeurope price.