Possible Medicare scam

My father who is enrolled in Medicare just called me and said that he kept getting phone calls about “added benefits” for Medicare and he didn’t have his Medicare card to give his ID number so he gave the person his SSN. He said he was trying to tell the guy on the phone that he is happy with his current plan and then the call dropped. He said he tried to call back and could not get through. Now he is freaking out because he thinks he got scammed. I’ve talked to him many times about not giving his SSN over the phone to anybody but he said it sounded legit. Is this a common scam? What can they do with his information now that they have it?