Yes, typically you will need to change your car insurance coverage from your previous state to your new state of residence once you have moved. There are a few reasons for this.
One reason to change insurance over is that most insurance companies do not want to continue to cover you under your old state's policy now that you and your vehicle are located in a different location thus with different risk factors, state insurance laws, etc. Where you drive and garage your vehicle makes a difference to an insurance company and once you leave the state they are authorized to do business in your insurance provider normally does not want to continue your coverage and will tell you obtain coverage in your new state.
The second reason is states normally require you to register, title and insure your car in their state once you become a resident. Residency laws differ from one state to another but in general you can be considered a resident if you take a job, buy a home, set up a bank account or even paying a utility bill.
While some states will accept out of state insurance coverage on your vehicle as we explained earlier many insurance providers will not want to continue your policy with your though once you move out of state. There are also many states that do not accept out of state insurance policies, one reason being that in-state insurers can submit to their database electronically and make it much easier for a state to keep track of who is and is not insured.
Some states that do not accept out of state insurance include New York and Arizona. The New York Department of Motor Vehicles site states that they do not accept any out-of-state insurance documents. For vehicles registered in New York State the liability coverage must be NYS insurance coverage, issued by a company authorized to do business in NYS and licensed by the NYS Insurance Department. Out-of-state insurance coverage of any type is NEVER acceptable or valid.
Arizona is another state that notes that out of state insurance is not accepted for Arizona registered cars. Here you must obtain auto insurance through a company that is authorized to do business in Arizona.
After you are move you may have a short period of time to get your license, registration, title and insurance changed over to your new state but it will need to be done and not delayed or else you may be penalized by your new state. You can contact your new state's Department of Motor Vehicles to get information on their state laws on residency and how long you have to change over your license, registration, etc.