It would be difficult. You'd first have to re-create your Federal return exactly as filed. And then start a state return. And then you would choose not to file the Federal return and to file the state return by printing and mailing (you won't be able to efile it.) If you end up needing to use a paid product, if you decide to continue, then you would have to pay for both the Federal and State returns when printing to file by mail.
Your state "may" have state filing on their state tax agency website. Some do.
See this FAQ for more info:
FAQ: Can I prepare and file just my state return using TurboTax?
