My question involves name change laws in the State of: GA

I live in GA but my son's biological father lives in OH, which is the state my son was born. We were never married and my son does not have his father's last name, he has my maiden name. His father does pay child support and is on the birth certificate but I was told by the state of OH that his father still has no legal rights to him because he was born out of wedlock. They also told me that in order to change his last name to my married name, it would cause for my son's father's consent: which is weird to me. But I just thought of hyphenating his last name to include my married name. Would I still need his father's consent to hypenate it?