Help in finding a biological father

I’m adopted, and I don’t know anything about my birth father. 4 years ago a social worker managed to contact my birth mother, but she wanted nothing to do with me (apparently she has a new family and she never told them). But they told me that the father didn’t even know about the pregnancy so he had no say about the situation. Apparently the plan was for him to never find out, therefore they don’t have his information. I recently found some documents with my birth mom’s name, so I tracked her Facebook, but she hasn’t used it since 2013, and there is no info whatsoever. I’m feel so stuck and I don’t know what there’s even left to do? I want to find him, but I feel like there is no way to do that, without her telling me who he is, but like I said- she doesn’t want anything to do with me.

Private investigators cost thousands of euros/dollars, and DNA test isn’t going to find a match 100%, and they are too expensive for just “maybe”.. please, if you have any ideas how to find him or contact her- let me know! Very appreciated♥️