The Social Security Death Index does go back to 1937 but there is little information in it prior to the late 50's when the Social Security Administration began automating their records.
The Social Security Death Index is generated from the U.S. Social Security Administration’s Death Master File. It only contains the records of deceased persons who possessed Social Security numbers and whose death had been reported to the SSA. In most cases a report of death was made in connection with Social Security death benefits.
If you find the person you are looking for you will also find a link to a form letter that you can send to the Social Security Administration to request a copy of their original application for a Social Security card. It will probably contain the mother's maiden name, father's name, occupation, birthplace and residence at the time of the application. You also have to send some money. The amount will be stated in the form letter. This request is covered under the Freedom of Information Act so the Social Security Administration must send you the information if you request it and send in the fee.