Yep that's not a problem, be sure to add the extra driver when you book the car hire so you don't get an additional driver charge at the local extortion rates. You will have to go with the additional driver to the original rental location along with your rental agreement, original ID and credit card etc, he will have to have his ID etc ready, and they will add him on for you. you'll be asked to initial the contract next to the written changes to prove you have made the amendments and not Alamo at random.
EDIT: When you pick up the car initially let them know that your additional driver will be arriving the next day and you will return to add him to the rental as per the booking / coupon.