[MODERATOR'S NOTE - Referenced post deleted as it was resurrecting a 5+ year old thread by a new user. Dialogue between users can continue below without the quoted text]

I JUST had warranty work done on my Volkswagen last week. Rented a car from Budget for 1 week. I originally put in that it would be returned at 1pm the day after I was supposed to get my car back..
My part came a day late so I ended up needing the car until 6pm that day instead. All I had to do was call and tell them. They didnt care at all. [/QUOTE]

If you picked up the original car after 6pm on the Pick Up Day, returning the car at 6pm (as oppose to 1pm) did not extend the amount of days (in terms of 24 hours) that the car is rented out.
Per OP, OP already had the car for 2 exact full days (48 hours), extension of a few hours will indeed means a new 3rd day being charged.

There are a few scenarios where extending a rental for a few hours won't get charged for a new full day.
1) The rate you paid for allows for more time than your original reservation
For example, extending a 5 days reservation by a few hours is going to cost nothing because a 5-days rental is priced the same as the weekly rate (7 days). So extending it for a few hours still falls in the weekly rate, may be a few extra dollars for taxes/fees that are per day.

2) The Corporate Rate / Special Rate allow for that kind of flexibility.
- American Express Platinum Card holders are allowed a 4-hour grace period at Hertz. So they can simply return the car a few hours (no more than 4) late and not get charged extra.
- A Corporate Rate with similar flexibility.

3) There are also some rental locations that have long standing business relationships with the local dealership/body shops and they allow customers referred by these dealership/shops a bit more flexibility, as a matter of mutual business benefit.