I searched the Irish Family History online database at ROOTSIRELAND.IE. It has abt 20M B/M/D records for most Irish counties. Records before 1863 are increasingly sparse: almost non-existant before 1800.
For John Moffatt B 1858 +/- 1 yr, somewhere in Ireland, with the parents' names unknown, I found 3 records. None are in Co. Tyrone.