I'm a bit suprised you have no info on him with already having a child with this man.
If this story is legit, you might be able to get the phone company of his former phone number to give you a name. You just have to talk to someone new
. Either that, or maybe a lawyer associated with the agency would be able to contact the phone company, get the name with a confidentiality agreement -meaning you dont get his name, just the agency- so they can contact him about his child. The other option is to contact a child support agency in your area and see if there is any way to track him down since if you dont know his name, he is probably not paying you support. And if he is paying support then maybe you wont be in this place of feeling like you have to place your current son for adoption. I would think a government agency like the child support division would be able to get his name from a phone number.