One hint: call the station directly, not central reservations. If you call central reservations, they'll reprice your rental at the current going rate for any changes (although some agents/supervisors may be able to override the rate change). If you call the station directly, they should be able to just adjust the time without making any changes to the rate, although if you no longer meet the actual rate rules (weekend pick-up restrictions, etc.), the rate may jump anyway unless the person changing your reservation (or renting the car) overrides it.

On a related but separate point, Avis doesn't play quite the games that some of the others do, but some rental companies are trying to fool each other when it comes to competitive rate shopping. That is, they are trying to make the rate lower than their competitors for customers who search for prices but hide that fact from their competitors so they don't lower the rate and engage in a rate war. Here are some things you can try to see if they lower your rate:

• Try searching for a pick-up time off of the top of an hour. hh:30 is good; hh:15 may be better, and something like hh:08 may be the best. Competitive rate shopping engines are usually programmed to check a standard time like 10:00am, so the savvy rental companies keep their 10:00am rates high but lower their 10:30 rates since real customers often aren't picking up exactly at the top of an hour.
• Try searching for a rental keep other than in exact 24-hour increments. If you're picking up at 4:00pm, don't schedule your drop-off at 4:00pm--try 3:30pm. This isn't want most rate shopping engines look for but it's the case for most real customers looking for rates, so the companies may price their 24-hour keeps higher and drop other time periods to be competitive.
• When setting weekly rates, most companies check the price of their competitors for straight 7-day rentals. Try pricing an 8-day rental. If it's less but you actually are keeping it for a week, just return it early. (Be careful when using a company that charges a penalty for early or late returns.)
• If you're picking up around the midnight hour, try checking prices both before and after midnight. Rates and availability are often set by the calendar day.

The rental companies are all trying to one-up each other and use (and are constantly discovering) countless other tricks to try to drive rates up while still underpricing their competitors for most real-world searches. That said, rental agency yield management is not nearly as developed as, say, airfare yield management, and there remains a large amount of guesswork and dart-throwing in the rate management departments of rental car companies.