Michael Joseph Connery (M J) left Limerick and immigrated to the United States in 1888. He was joined by his 16 year old sister Cath on the ship Alaska which docked in New York on July 30,1888. He was one of 11 children which might explain his choice to immigrate.
I have explored MJ's siblings and their descendants, but have not studied the surname Connery. The 1901 census for Ireland shows 450 entries for the surname Connery with 129 Connerys living in County Limerick and 10 in Kilfinnane which is MJ's birthplace. The 1911 census of Ireland shows 467 entries with 114 in Limerick and 20 entries for Kilfinnane. In contrast, the 1900 Census of the United States there were 79 Connerys with 7 Connerys in Illinois and 3 living in Cook County, Illinois. The 1910 Census lists 71 Connerys with 7 in Illinois and 6 in Cook County. I only looked for Connerys who were born in Ireland. New York, Massachusetts, and Illinois had the highest number of Connerys listed in both the 1900 and 1910 census.
I will need to further explore the name Connery. There must be more to find.