I don't know if you have checked out Griffiths, but it gives 2 Harrigans and 3 Herrigans in Mayo in mid 19th cent, in addition the 1901 census has some
That would give you the parish, and you could then do some local work by contacting priest or local historian in that area.
If you want to go the whole hog you get dna tests done from male Aus Harrigan and compare with any Irish Harrigans that you can get to do the test (it is possible, I have done this with my own family, including tracing descendants of a convict in Australia)