Áine's suggestion is by far the best. Irish started to settle in Texas at least by the 1820s. Put "Irish settlement" in the search box at Texas Online website and over 1400 items come up. You will have to be patient with this website, errors in making connection pop up from time to time.
The 1850 census lists 1403 Irish in Texas in 1850 and 3480 in 1860. In the Mexican days, "whites" were lumped together in counts.
The common ballads and songs brought over from the British Isles are a rather poor place to start, since it is very difficult to demonstrate whether their origin is English, Irish or in many cases Scots. Examine the threads on these songs in Mudcat in detail and you will see just how difficult it is; also apparent are the myths that have become associated with them.
The American-Irish Historical Society Journal may be found at a nearby university; much data there. Also see Flannery, "The Irish Texans."
You also might look into "The Irish Channel" in New Orleans.
Many Irish came during the Civil War period; many of those who landed at northern ports fought for the Union and those who landed at southern ports fought for the Confederacy. I had a great grandfather who came over in 1860 and fought for the Union, but after the war he went to New Mexico.