John Rand Mills of Ballysheehan

I recently requested records from the Warren County Muncipical Center for my great-great-great grandfather with one document specifically outlining where he was born:

John Mills...doth despose and say that he is of the age of thirty three years [putting his birth date in 1805] that he was born in the Parish of Ballyshehann and County of Tipparary in that part of the Kingdom of Great Britain called Ireland that he embarked at Waterford [Ireland] in the month of July as he believes in the year 1835 and arrived at the city of New York in the State of NY in the month of Augst and following that he has since resided in the state of New York and county of Warren and still resides there.

Tax assessments from 1827 and 1828 in Ballysheehan, Ireland listing a "John Mills" presumably would be him, as would a "John Mills" listed in Tithe Applotments of Ballysheehan Civil Parish in 1827. While he is alive, the 1850, 1855, 1860, 1865, 1870, and 1875 censuses, while living in Warren County, NY, clearly indicate that he was born in Ireland. The same is the case of censuses of his children, in later years, listing their father as living in Ireland. What I've found so far is attached as a PDF.

What suggestions do you have for where I should look next? I've already looked at many online sources and am basically up against a brick wall as it stands right now. Thanks.