Have you checked with the Register of Deeds in Cumberland County to see if your grandfather filed a delayed birth certificate? It would require proof of the parents' names.

If he filed for Social Security, you can get his application at:

https://secure.ssa.gov/apps9/eFOIA-FEWeb/internet/main.jsp

Even if he is not in the Social Security Death Index, he may still have applied if he was still living when SS was started.