You didnt' include location or time frame of his birth. In a few cases, it could be online.
Not knowing his bio father means either the parents were divorced, or he was born out of wedlock. Look for the marriage record of mom and step father..and what name she used to get married. If her maiden name, then she probably was not married at the time of his birth. Look for a birth under the maiden name. If she used something other than her maiden name.. then it is likely she was married before. Look for that marriage. Then look under that name.
Another option (depends on YEAR OF BIRTH) IS to look for mom in the census. You may find dad at (say) age 5, before she married the stepfather.
Taking the name of a stepfather was often done informally, without a legal adoption. This means, the original birth record was not changed. Adoptions legally (as we know them today) generally will amend the birth record to show the adoptive parent.
Location matters. www.familysearch.org for example, has full birth records for 1903 to about 1935. Calif birth index is also online there, although not sure of the inclusive dates.
You are welcome to email me though my profile with the details, and I'll see if I can't find it for you.
edit
meant to say, the birth records for Texas 1903 to 1935ish are online. Not all states have this.