I agree that the social would have to match the name for the federal to not get rejected.   However, the state doesn't always work that way.  

If either fed/state was filed incorrectly.   I would send the amended return with the explanation of "Clerical Error" and state the original filed social and the correct one - enclose an "As originally filed" return copy as well as an "As amended" return copy with the 1040-X   or other state Amended form. 

Seem like amended would have to be paper filed as Intuit software wont let you alter the SS# once it has been transmitted.