The website of the Department of Motor Vehicles in your new state will contain all the legal requirements for insurance and vehicle registration. Depending on the state, you could have 30 to 90 days to change your insurance, driver's license and registration. You could face a fine for not making the change by the deadline. During your progressive quote, you won't be able to select car insurance coverages that don't apply to you.
Every state requires auto insurance companies to declare how they calculate rates to customers, and insurers cannot deviate from these presented rates. In that case, you would need two car insurance policies, one for the vehicle that is kept overnight in Connecticut and a policy from another state in New York for the vehicle you keep at work.