You need insurance on both cars. Your insurance on the new car would not extend to the older car, each needs their own coverage. You can have them both on the same car insurance policy, a multiple car policy, which should give you a discount however you cannot just have one policy on the newer car and try to use that proof of insurance for the older vehicle.
Simply, you can have liability coverage on the old car and the new car, but have full coverage (physical damage also referred to as comprehensive and collision coverage) on the new car. All of this would be on the same policy so you are properly protected and get all the available discounts.
There are a few reasons that you need insurance on each car. One is that most states require that you have insurance on a vehicle when registering the car. Thus if your older car is without its own insurance coverage than your state will be informed and could penalize you. The penalties for a lapse of insurance vary by state but can include fines, registration suspension and even your driver's license being suspended.
Insurance on your newer car will not be accepted by your state's Department of Motor Vehicles (DMV) for registration purposes nor would it be valid proof of insurance to show to law enforcement since it does not cover that specific car. Your insurance coverages typically cover the vehicle listed on the policy and some portions of the policy (such as Liability) may follow you when you drive other cars, not owned by you, though only as secondary coverage.
Insurance coverage follows you in some ways but not in the way you are asking with regards to owning 2 cars. Your personal auto insurance will usually follow in a situation such as you are borrowing a friend's vehicle and are in an accident. Typically your Liability coverage will follow you in this scenario but only as excess or secondary coverage to be used after the car owner's (friend's) primary coverage. Car insurance follows the vehicle first and driver second so the car owner needs to have insurance on the vehicle with proof of insurance that lists that vehicle on it.
With your second car there is no primary insurance coverage on it, so it would be considered an uninsured vehicle and you could be cited for driving without insurance if you were caught operating it.
You can get free auto insurance quotes for both of your vehicles here.