I sometimes give advice to adoptees who are in search of an unwed father whose name is unknown, but without feedback from the adoptees, I don't know if my advice is good.
There is a possibility that the birth father contributed a few bucks toward the maternity care---the hospital bill, If so, his name might be on the financial record if it still exists.
If the pregnant woman sought welfare benefits, an agency may have filed a paternity suit against the alfa (alleged father). I wouldn't know how to look that up at the courthouse even if I knew his name.
Prior to the Supreme Court case known as Stanley vs. Illinois, there was little incentive for social workers to pressure the pregnant women into revealing the names of their lovers---it would just be one more person to hunt down to ask to sign a surrender form. But the adoption agency might know the father's name.