help with the adoption process

What is the name of your state?Texas
I had my daughter my senior year in high school and at the time her biological father wanted nothing to do with her. I gave her my last name and left the father on the birth certificate blank. He soon after she was born had me in court signing over his parental rights. She is now 9 years old and her step father and I have been married for 4 years now. Her biological father died a year ago in December. I want to have her name changed on the birth certificate. Do I need to file for an adoption? Why can't I jus t have her name changed through the social security office since there isn't a father on the certificate. Please help me in this matter. She now has a half brother and a step sister that have my married name and this makes the difference very obvious to her.
Thank you.

If the biofather's parental rights were legally terminated in court, why not just do a legal name change? That would be a fairly simple matter, legally. No one but you and she need to agree about it. Biodad is out of the picture, and this would solve the different name problem.