Mena Family - Research on Illegitimate Child

Hello, everyone. I'm researching the Mena family. Our family comes from Lagos de Moreno, Jalisco. Specifically, I'm mainly interested in finding the ancestry of my bisabuelo Jose Jesus Mena, born in Lagos de Moreno in June 25, 1897. His mother was named Melquiades Gonzales born around 1872.
The major problem I am having is that he was illegitimate. The civil registration record of his birth does not list his father's full name or his paternal grandparents. The birth record indicates that his birth name was Jesus Gonzales, but all records following immigration to the United States in 1916 use his father's last name - Mena. In his naturalization papers, he says that his father's name was Manuel Mena.
I've looked up several Manuel Mena's who were born in Lagos de Moreno or la Union de San Antonio around the same time as his mother. There are a few possible contenders, but I don't know how to move ahead.
One possible clue is that Jose Jesus Mena was remarkably tall (around 6'2") and had very fine Spanish features. I've wondered if I could research photos of Mena families if I could make a probable guess.
Has anyone else had any luck researching illegitimate children or have any advice?
Thank you for your help in advance.