There's no record of a Hugh H. or Hugh A. Logan in the 19th Arkansas Infantry, nor in any other Arkansas regiment. It is possible that this man served late in the war -- a period for which there are no surviving muster rolls.
It's also a possibility that a VA marker was issued for his grave in error. VA markers are absurdly easy to obtain, and many are issued in error. The VA relies on the certification of the applicant that the deceased served in the Civil War, and does not independently verify the information. Unfortunately, far too many markers have been issued, based on sloppy research, so I never take the issuance of a VA civil war service marker as proof of anything, unless I can independently verify the service from other sources.
It's a shame that we've come to that point.