It depends on who you're hiring from. If you're hiring from a small, local US hire car company, you might well be out of luck. If you're hiring from somewhere big, they should be able to help
Avis UK have a handy page on hiring a car without all of your UK driving licence, which covers what you can do depending on where in the world you are. They say that for Avis US locations:
Renting from Avis in the USA
When renting in the USA, you can rent a car with your photo card licence only, or with your paper counterpart licence and your passport. If you have neither part you cannot hire a car from an Avis office in the USA
Hopefully other larger companies will be able to do the same. Somewhere small is most likely to only be used to photocard driving licences, so may well not be able to satisfy themselves and their insurers with only the paper part and a passport.
In the UK, you can (normally for an extra fee) have the hire car company phone up the DVLA, and get confirmation from the DVLA that you hold a valid licence without too many points. That only works when the DVLA are open, and requires that the hire car company know about how to check with the DVLA. Maybe you could find a US hire car office that knew about how to do it, but it's not that likely, and you'd struggle with the time overlap of when the hire car place is open and when the DVLA call centre is open. You could always speak to the head office of the hire car place to see if they can set this up in advance though, you might get lucky but don't count on it!