I haven't been able to connect any of the Morehead family trees to William Morehead, who was the father of Henry Harrison Morehead.
I see two censuses (1850 and 1860), where he is shown as married to Judith Barker (I feel confident on this connection because he names his son Thomas Barker). He is noted as "deaf" in the comments on both census documents.
Henry Harrison's death certificate has the name spelled Moorehead, for both himself and his father (this is a common spelling variation).
Henry Harrison's 1900 and 1910 census documents list his father's birthplace as Ireland, but documentation that I've believed to connect them to Alabama (with mother Julia/Judith and next door neighbors to Barkers), but this "William" indicates his birthplace is South Carolina, not Ireland. Whereas, Henry Harrison Morehead indicates he was born in South Carolina but ended up in Texas.