How exactly do you plan on using these hashes to ensure uniqueness if you are going to hash them with a patient specific salt? If you want to know if you have a patient with SSN X you would have to go through every patient record computing however you plan on doing it with their salt plus the query SSN (you could make it not so horrible by ordering the checks by similar names/birthdays/whatever).
Since you are talking about generating an identifier from the SSN and using that instead there is no benefit to using the SSN in the first place. Generate a UUID and use that instead.