Looking for Vernon from the Birmingham Alabama Area

My name is Brenda G. VERNON White. Both my father and Mother are deceased. I wish daily that I had started this when they were alive to help with information.
My fathers name is Lloyd Robert VERNON (b1932 d 2004), he was born in Birmingham Al and had no siblings. His mothers name is Mary Addie McDaniel (b 1912 d. abt 1959.) She was born and died in Birmingham AL. His fathers name was also Lloyd Robert Vernon (maybe spelled with one L rather than two. On my fathers birth certificate, it states that his mother was 20 yrs of age and his father was 22.
The catch is that they were never married. The senior Loyd Vernon later married someone else and as I understand the story they had at least one daughter. My father and grandfather never associated and as a result, myself and my siblings never met him and know nothing of him. I am pretty certain that he remained in the Birmingham area and may have passed away in Jefferson county.