Typically what happens when you rent a car is you will get a list of all the applicable charges on the rental contract. It will say something about the weekly, daily and hourly rate. I don't think you'll have to change the actual reservation now and risk losing your low rate. You can just tell them at the counter when you pick up the car that you will be keeping the car an additional day and you will pay whatever the daily rate was per your original quote. Your best bet though to confirm this is to call the rental company.
One trip when we were in Florida, my wife had problems with her ears and we had to stay for a couple of extra days. I just called the rental company's local office and told them I was keeping the car a few more days and I only had to pay the additional daily cost as quoted on the contract. It didn't effect my discounted weekly rate at all.
But again your best bet would be to check with the company and verify if this is correct or not.