If I think an estimate for damages to my car following an accident is unfair, can I take my car to get another insurance estimate?

Absolutely, in fact we encourage you to do so. By you going to get another estimate, one, you're going to have sound mind that your car is being repaired properly, and also it's going to allow you to negotiate the full and fair settlement for the repairs to your vehicle with the insurance company. Now, just because you take your vehicle to somebody else who may say that your vehicle is damaged further, that does not necessarily mean that the insurance company is going to pay for that. Therefore, I suggest that you contact an attorney to help you through this matter.