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This particular record has no family information. That is not unusual - in fact it is common. At the time, the government saw little need to record that sort of information. This man's only card indicates that he enlisted in this militia unit in Andalusia, Covington County, Alabama. This was a 90 day unit only. The only man I can find residing in Covington County by that name in 1860 is a William P Williamson shown as a farm laborer, aged 21, living in the household of a grocer named Andrew J Fletcher.
Here is what an inspecting officer had to say about the 4th Ala. Militia: "This Regt. is composed of 'Ninety days Men' who have armed & equipped themselves. Have been employed in the City of Mobile mainly as a police force and have had but little opportunity for improvement in Discipline Instruction."
The records of Capt. Gantt's company indicate that many of the men did indeed enlist in Co. I, 40th Alabama Inf. Regt. However, as Michael indicated, Williamson does not appear to have been one of them. He is also not found in other units commonly associated with Covington County. If his ties were mainly to another county, perhaps some further research would find him in another unit. However, unless he died during the war and his family made some claim for any pay due to him, it is doubtful that you will find any information on his parents in his war records.