Will my insurance company pay for storage fees?

I have uninsured motorist coverage and I was rear-ended by a hit and run driver and my vehicle was towed to a storage facility because the vehicle was inoperable. It took a little over a week for me to receive the police report and now the insurance company is trying to find the other driver. I was told by the adjuster that they would only be responsible for the storage fees once they determine that it is a hit and run, meaning I will be responsible for the fees up until they exhaust their efforts in locating the person who hit me. First off, I can not afford to pay the tow fees and secondly I live in an apartment complex that is very strict when it comes to vehicles and they will have any inoperable vehicles towed. What options do I have? Will I really be responsible for the tow fees?