I filed my federal return with another service but would like to file my state return with TurboTax. How would that work?

When you use TurboTax the information flows from the federal return to the state return.   You cannot use TT to prepare just a "stand alone" state return.   You could try using the state's tax site to prepare your state return.  You will need some information from the federal return you filed.

https://ttlc.intuit.com/community/state-taxes/help/how-do-i-contact-my-state-department-of-revenue/00/25573

You can only efile state if Federal was efiled and accepted from the same account.  So you will have to print and mail state.

If you didn't use Turbo Tax for federal, There is a way but you first have to fill in the federal and pay for federal for it to flow to state.  You will have to mail state.  See,

https://ttlc.intuit.com/community/filing-and-paying-taxes/help/can-i-prepare-and-file-just-my-state-return-using-turbotax/00/26408