As he seems to be the only one of that name and age from those parts, have you wondered whether in 1881 he might be the Edward Hamilton, 21, born Usk Monmouth (dittoed from above entry)
In barracks at Brecknock, a private in the Royal South Wales Borderers Militia.
There is also a record of an army deserter called Edward Hamilton
Deserted 16 May 1881. Age 21, born Pontypool, the index says he deserted at Monmouth. Possibly from the Monmouthshire Regiment (I haven't seen original)