This is the reason why it is worth checking with your insurance company about how your car insurance is affected by your change of address.
For example auto theft is a much more serious risk in certain neighborhoods than it is in others. So if your insurance cover was taken in an area where auto thefts are a very low risk, you cover probably did not address this issue adequately.
An insurance company will tend to have all the latest information and statistics on all these risk factors. So by consulting with your insurer you will also be made aware of the new dangers and higher risks you face in your new neighborhood.
Another factor worth considering is that your new neighborhood may be much closer to your work place meaning that you will probably use your car less. This is a factor that affects your car insurance because it is assumed the more you use your car the more opportunities there will be for accidents or possible accidents.
You should be able to review the change of circumstances in relation to your current car insurance and make a quick decision whether you honestly still have adequate car insurance or not.
There is no harm in negotiating a new car insurance deal even when your old one has not expired, simply based on your change of circumstances.