Am I allowed tooffer evidence to the adjusterthat my car is worth more than what he or she hasoffered?

Sources of extra value are upgrades you've had done on your car, or accessories you've purchased for the vehicle.  Perhaps you've recently outfitted your car with four new tires.  The adjuster will ask for receipts or other proof of these improvements, and won't increase the offer without that proof.

Keep in mind that the adjuster does not have an obligation to compensate you for general upkeep of your vehicle.  Normal maintenance, replacement of tires when you have a flat tire will not earn extra consideration.

The rules and practices are often ambiguous.  Individual insurance companies handle property damage in widely differing ways.  What may be the rule for one company will be the exception for another.