Ladies, I need help.

My daughter's biological father left us when I was 15 weeks pregnant claiming the baby wasn't his and what not. My daughter is now 4 months and he hasn't done anything. It's like he never existed in our lives. The baby is under my last names I don't get child support because I don't wanna force him into it.

My question is, when I get married. Can my daughter have my husband's last name? I've read about him being able to adopt the child if the dad gives up his rights.. But I don't think he has rights because he isn't on the certificate nor does she have his last names. I believe a paternity test would have to happen for him to have rights?