How possible is it to negotiate a larger settlement with a third-party insurance company?

My car was hit while parked last year. The car is a total loss. The driver was driving a rental car, was apparently not personally insured, and hit three cars (including mine) while going around a corner. Two of the parked cars (including mine) were totaled, and the third sustained minor damage. Due to my own stupidity and absent-mindedness, my insurance had just lapsed, and so I have been working with and waiting on the company through which the rental company insured the car.

From the outset, the claims manager told me that the policy had a $25,000 maximum, which would have to be spread over the three damaged cars. It took a long time to figure out the settlement, because they had a hard time getting hold of the other drivers. I have now received a settlement offer (or, they sent me a check in the mail, which I assume is the settlement offer as they have sent me nothing else official), and it is a few thousand under what the average market value of the car is.

I have several questions I am hoping to get some insight/help with:

• I understand that people have been successful in negotiating larger settlements with their own insurance companies, but do I have any way to do that with this company? Or is that really only something that happens with your own insurance company that wants to keep you as a client?
• From reading this sub, my understanding is that the way to negotiate would be to find local/regional listings to establish the fair market value and then submit that information to the claims agent. Is that correct? Is that something I should do by email or by phone? Or does it matter?
• If working with the insurance company is a dead-end (or non-starter) in my case, would it be wise to get an attorney to deal with the rental car company and/or the driver?
• I do not have a driveway and had to move the car off the street. Is it common/standard for insurance to cover any costs for storage? (Fortunately I had a friend who allowed me to use their driveway, so I have not incurred any actual storage costs, but I am wondering if the issue of storage is worth mentioning when discussing the offer with the company. The company also told me at the outset that I would need to keep the car until the settlement or I might not be eligible for anything, and so that has kept me from selling/scrapping/trading it in.)

I read through the rules and FAQ and did not see these issues mentioned, so I hope these are appropriate questions to ask in the sub, but if not and the thread must be deleted I understand. Meanwhile, thank you for any and all help/insight you may be able to provide.