Trying with difficulty to find birth county of my great grandfather Martin Dermody who on the English Census of 1881 came from Dunmore, Ireland. In 1870 at the age of 23 yrs, he married Mary Connelly in the Roman Catholic chapel of Ethaldreda Hodgley, Rugeley in the district of Lichfield, Staffordshire. His marriage certificate gave his father's name as Patrick. Mary's father's name was given as John.
I have found no English census records before 1871 with Martin's details (or his father's) and can only assume he came over from Ireland between 1861 and 1870. My initial and unqualified searches narrowed his birth county down to Galway or Tipperary, but then I came across the 1881 census record with his birth place as Dunmore. I understand there are about 15 townlands in Ireland with that name place.

Can anyone please give advice on my best way forward to finding Martin's birthplace. Ireland seems so hard to research with its fragmented and incomplete databases.