I have completed Y-DNA37 testing with a Gavin family member from Mountbellew and we are a close match with only 2 deviations in almost 200 years. This relates us back about 8 to 10 or so generations or so. By the paper trail I have constructed, we knew it was more than 6 generations anyway. This confirms my lineage back to the area. I was able to find a marriage record from 1851 for my 2x great-grandfather Patrick Gavin (1826) that his father was also named Patrick. Possibly one of the P. Gavins on the Ballinlass Memorial. All the time frames seem to fit.
Also found some unknown relatives from my Great-grandmother Barry's side that had done their families genealogy (and mine). I would recommend using DNA testing to confirm or disprove your hunches. Watch for sale prices in November!