Here's a possibility. However, there are quite a few W. H. Walkers out here.

W. H. H. Walker
Residence was not listed;
Enlisted on 9/2/1862 as a Private.
On 9/2/1862 he mustered into "E" Co. KY 42nd Infantry
He was Mustered Out on 10/2/1862

Sources used by Historical Data Systems, Inc.:
- Report of the Adjutant General of the State of Kentucky
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