Regardless of the type of insurance claim that you are talking about, the one thing that will make the process move as quickly as possible is good communication. There is a significant amount of communication that has to happen to move a claim to completion. Make sure that the process is never waiting on you.
First, contact the insurance company as soon as possible. The sooner you get the claim started the sooner it will finish.
Second, be responsive. Give the insurance adjuster your cell phone and your email address. The easier it is to contact you when they need something the faster you will be able to respond. Don’t wait for days to call back. Don’t take a week to get back to them with a copy of the towing receipt. Are you in a hurry to have this claim processed or not?
Third - drive the process. Every single time you talk to the adjuster ask them what is the next step in the process and when they expect to be done with that step. Then on that day send the adjuster an email or call asking for status. Most adjusters are handling multiple claims. Make sure your claim is the one they want to get done next.
It is not at all uncommon to have an adjuster complain to me that they can’t get in touch with one of my insureds and at the same time have my insured complain that the insurance company is dragging their feet. Feel free to talk to your agent if you are not sure what you should be doing. Communication is hard. If you don’t believe it just ask my wife.