I believe you are referring to diminished value. Diminished value (DV) is a term used, in regards to auto insurance, to mean the difference between what your car was worth before and after a car accident. If you can show that due to this accident to your new car will suffer from diminished value you might be able to negotiate with the other party's insurance company for it.
How successful you will be with obtaining money for DV largely depends on what state you live in. Several states, including Georgia, have had judges rule that insurers must compensate insureds for any real loss of value from an accident. There are also many states, such as Texas, that believe that DV is not necessarily true if the car is repaired properly and thus does not allow for DV.
You should contact the at-fault party's insurance company or perhaps a lawyer to see if diminished value is allowable in your state to be negotiated for. For more insurance terms and definitions you can visit our Learning Center.