Hello again Carol,
You may be looking in the wrong place for 1860s Carrickmore Whites.
The 1901 Irish Census for Tyrone has been indexed on micofiche by a worthy researcher, and my quick search this morning indicates that there were only two Whites/Whytes in Termonmagurk in 1901. One was a Sarah White, a 50 yr old widow living in Copney townland, who may or may not be one of your bunch. However the other individual is, I'm sure, one of yours. She was an Emilie A. White, a 38 yr old C.o.I. spinster living in Athenree townland, but the curious thing about her is that her place of birth is recorded as 'England'.
You may get a lead at 'findmypast.com', the England GRO website.
I'm not suggesting that the Whites were English, at least not 19th century English, but they may have left Ulster, spent a while in England and then returned. However, Carrickmore may not be where they came from originally. The 1860s Household List (Griffith's) records only one 'White' (Michael) living in Cooley, a townland some distance from Athenree/Carrickmore.
What I'm saying is - if your are looking for a George White then be prepared to cast your net a little further than just Termonmagurk. Every Irish family had a faction who emigrated to other continents, but those that stayed weren't entirely sedentary. In the space of 80 years my own Ellisons migrated from Sixmilecross, through Clogherny, Killeeshil and Carnteel Parishes, almost within a short walk of Co. Monaghan.
Brian