If your vehicle is involved in an accident, you can choose whether the fault lies with you or it's not your fault. If you are at fault, you can get a rental car with your own insurance company if you have rental car insurance.
If you are innocent, you can get a rental car with driver's insurance. Some state laws require insurance companies to pay the offending driver a "reasonable transportation fee" or "reasonable replacement vehicle rental fee" when repairing your damaged vehicle. You can also get more information about how to rent car after accident online.
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Guilty driver's insurance usually doesn't allow rental cars until they can talk to the insured and confirm that he or she is really at fault. This can be a problem if the insured cannot be contacted by you or if he or she is lying about the accident.
Most companies pay a fixed amount, such as $20 per day. However, if there are special circumstances that require a vehicle similar to your damaged vehicle, report this to the insurance company to see if this cost is covered.
You need a car – any car – so you will argue about the same car as yours, but you need to get back on the road soon.
You have kids to take to soccer practice and you have to go to work in the morning, so "basically staying" may not be the best approach. You have to choose your battle and right now your main need is to get another phone.