It is possible if you re-create your federal return in the TurboTax software or TurboTax Online but don't file it. Then prepare and file your state return using TurboTax. You'll need to paper-file your state return, as state returns cannot be e-filed separately. But you might have to pay for both federal and state returns, even though you won't be filing your federal if you do not qualify to use the TurboTax Free Edition.
Please see this FAQ for details:
https://ttlc.intuit.com/replies/4232481
