If they towed it, you'll be looking at the tow and storage costs in addition to salvage value.

Quite some time ago, my car was parked in front of my sister and bil's house. While we were out at the races in Dacono for the evening, a drunk driver plowed into it and then another as she tried to flee. My car was never towed from the scene, (but we did have to move it out of the yard and into the driveway).
After what seemed like too long a period, the insurance company finally offered up a check for totalling it.
My bil inquired about buying it for salvage because he wanted to make a demo derby car out of it. After hearing their asking price, he presented them a bill for storage, since it had sat on his property since the incident.
The bill was more than the asking price, so they signed over the title.
Side note:
Rear axle got snapped early in the first derby, so after a trip to the junkyard for a replacement, it made a second derby appearance where he lasted a bit longer, but still got his ass handed to him.

TLDR; if you maintain possession of the vehicle from early on, you have leverage.

Sent from somewhere...