Not so easily done!
If a passenger misses a connection, the airline will arrange for alternative transportation on a later flight. It's the passenger's responsibility to check with an airline agent to determine what the alternative schedule is. If the passenger misses that onward flight (ATL-SAV in this case), the airline will treat the passenger as a no-show and will cancel the remainder of the itinerary, including the trip back to Seattle!
DL has flights from ATL to SAV about every 90 minutes or so during the day. So it's likely that DL will be able to get you there fairly quickly. However, even if the delay is overnight, don't rent a car and disappear without getting airline approval to skip that ATL-SAV flight.