Yes, if you have an insurance policy you're covered throughout the United States. Moving to a different state temporarily may require changing coverage, depending on the amount of time a state allows residency with out-of-state insurance.
Moving for the long-term definitely requires switching insurance, although the time to do so can be somewhat variable. For example, a state may technically require changing after 90 days, but if you have two full months left on your current policy it would seem silly to switch prematurely. However, the "grey" period in between could prove problematic in the event of an incident.
