How can we extend the car rental liability of an insurance company until they provide a check for a new car?

Think about what you are attempting to do: make an insurance co provide you a rental car until you agree with them on value. May take a lawsuit and a couple of years in court. You are entitled to the fair market value of your car and your opinion may not matter and what you want from them may not have any relation to FMV. Buy you a junker or rent a car for yourself and sue the rascals if the company is trying to low ball you.

Welcome to the wonderful world of insurance settling. Be tough with them. The problem is that they know that their liability is limited to x dollars, whatever x turns out to be. If you were injured, on the other hand.

They have a legal obligation to pay for the use of a rental car for a reasonable period of time to allow you to obtain another vehicle. There is no fixed definition of "reasonable." Certainly it can be argued that the period of time only starts once they make a reasonable settlement value offer for the car, which is in pat what they are saying. I do not know whether their offer was reasonable or not (normally it would be less than the cost of replacing the vehicle and just what its fair market value would be if you wanted to sell it). Just because it was an offer does not make it reasonable. The requirement that the rental period end two days after any offer is not reasonable. They may be referring to their contract with their own insured (what they have to pay their own insureds) but that contract was not signed by you and is not binding on you. Send them a demand letter for the full fair market value of the car and that they pay a fair market rental for your car (does not matter what you actually paid for the rental if below that or even if you rented a car you do not have to show them a receipt) for a reasonable period of time once you get the settlement money until you can buy another vehicle. Two days is not a reasonable period of time to do so, as you do not know what you can afford as a replacement vehicle until they settle with you and you actually get the money form them. If you take a firm but calm enough position in the letter and in fact they will likely bend at least a little. Paying an attorney to write a demand letter might be worth it depending upon the reasonable?difference in what your demand and their offer is if the difference is a few hundred dollars it is not worthwhile.

Make a collision claim on your own policy and let your insurance company worry about getting their money back.

You sue the other driver to get your money back for the rental after the cut off date. However, if you were also injured, then in that suit, also make your claims for injuries and damages. You may need to consult with a local lawyer.