The deciding factor in this scenario is that insurance companies are regulated at the state level. This means that moving to a new state, even one where your current insurance company operates, will require you to get a new insurance policy.
Insurance rules change from one state to another, and the state required minimum coverage is different as well. Similarly, the type of system used for insurance may be different, with many states using a tort system and other preferring the use of no-fault coverage that precludes court involvement.
Most states will give you a grace period of up to 30 days to get your registration, drive license and insurance information updated after moving to a new state. After that grace period has expired, you are subject to be ticketed or fined if you are stopped with the insurance information from another state.
On the other hand, if you are traveling through another state, your current insurance will probably cover you in case of an accident. Because state laws and insurance requirements vary, you should contact your insurance company before to traveling to make sure that your coverage will be acceptable. In a situation where your current coverage is too low, the insurance company can offer a rider on your policy that extends your coverage for the necessary amount for a limited time.