Typically there is not a grace period for car insurance. State laws and insurance companies' guidelines vary; if you do not pay before your due date than your auto insurance policy will terminate and you will be without coverage. This applies whether you're insuring a brand new car, a recently purchased used car, or switching insurance companies.
For example Nevada's Department of Motor Vehicles states very firmly that NV has no grace period in regards to car insurance. A one day lapse in your insurance coverage in Nevada will result in a possible suspension of your registration. A fee of $250 will have to be paid to reinstate your registration privileges.
In Nebraska there is a statute (NE Statute 44-516) that requires a company to give 10 days notice before cancelling a policy due to non-payment the NE Department of Insurance notes. This is typical in most states for late payments, but not for policy dates.
If your policy has canceled and you want to have it reinstated after a lapse and there are no specific insurance laws your state regarding this than it is up to the discretion of the insurance company according to the rates and guidelines that they have filed with the state.
You can check with your State's Insurance Regulator to determine the specific rules that apply in your state for how long after the due date your car insurance can be cancelled if the premium is not paid and if the state has any grace period that auto insurance providers must grant you.