No, because most states piggyback off the federal return (using federal AGI as a basis in state calculations).  So if you file a state return attesting to a federal AGI that doesn't exist yet -- and can't be verified with IRS -- your state may kick back the return.

If the purpose of wanting to file state is to get the refund, and then you pay IRS what you owe them, you can accomplish the same goal by e-filing both returns and then simply waiting until the state sends your refund before paying the IRS.  You have until April 18 to pay; you don't have to actually pay with your tax return.  Obviously, the risk here is that the state holds your refund for some reason, in which case you'll still need to pay IRS on time to avoid penalty & interest.