Are you looking for descendents or ancestors? If ancestors, then your next step would be to get the death certificate and obituary of your grandfather that you mentioned. Typically obits tell you where his brothers and sisters are living at the time of his death, death certificates tell you who his parents were and possibly where they were born.

Also you should obtain his Social Security Card application file which will confirm some of this info, or give it to you in another way, and also confirm his birthplace. Let us know how you proceed.