I searched the Alabama Death Index and found several entries for William Cox in 1956 (none in 1957 or 1958).
William Cox Death Date: 06 Jun 1956 Death County: Jefferson Volume: 25Certificate: 12079Roll: 5
William A Cox Death Date: 25 May 1956 Death County: Chilton Volume: 18Certificate: 8929Roll: 5
William D Cox Death Date: 13 May 1956 Death County: Jefferson Volume: 20Certificate: 9618Roll: 5
William A Cox, Sr Death Date: 27 Jan 1955 Death County: Montgomery Volume: 4Certificate: 1579Roll: 5
You might want to order these death certificates and see if one of these men looks like your William Cox.The death certificate should indicate the names of the parents of the deceased.(www.vitalrec.com has more information about ordering certificates)
Good luck! Karen
PS Have you found William in the censuses yet to see if either of his parents ends up living with him?