So an aide-de-camp wouldn't have enlistment records? The second application says his rank was Captain.
Officers were entitled to a confidential assistant, known as an aide-de-camp. The aide's duties included writing and delivering orders. This was a position of responsibility since he knew troop positions and where officer quarters were located. Many times, aide-de-camps were family or trustworthy friends.
Aide-de-camps moved with the general assigned to wherever he went, even if he was transferred. So, it makes sense that Hugh followed Gen. Whitaker to the IV Corps (or "4th Corps", as described on the wife's pension requests).
I'm not seeing any obvious army records for Hugh, unless he served under an alias? Perhaps he pivoted into an aide-de-camp role due to his clerical background -- and
who he knew. (I note his occupation on the 1860 census was "clerk").
Have you checked to see if Fold3 have any records for Hugh?
Apparently...the Compiled Service Records (Fold3.com) of
volunteer Union soldiers are available online.
"In the future, these records will be made available at no charge through the National Archives website".