Blythe was serving in Europe with the US Army until 5 months before Bill's birth, at full term. He was also married to someone else in Texas, and had children, (both of whom have prison records) at the time. There is no solid evidence that Blythe was actually married to Virginia Kelly, Bill's Mom. Old Hot Springs hands think that Bill's Uncle Ray, the town fixer, arranged Blythe's marriage or the records, possibly posthumously, to Virginia. (She is the Mom. That's certain, at least.)
Bill Clinton either has no idea exactly who his Dad was, or does not wish it known. Neither does the Blythe family.