Dear Mary, I just saw your reply and have many ideas which I can't pursue tonight as it is 'silly o'clock' here so I am going to list them out in case I forget them.
This link lists Greely names http://www.kancoll.org/books/andreas_ne/greeley/greename.html Are the Whites among them?
were the Whites members of the local Catholic Association?
Did Bishop O'Connor write a memoir?
Is it possible that copies of the Greely County Tribune survive or the O'Connor Democrat?
Did the records of the local undertaker survive?
How about school records? Sometimes children did projects about their families and samples survive.
If Thomas worked for the railroad or the mines, there may be union records. Sometimes men worked with other men from their home place?
Do you know anything about the Irish policeman who helped Mary Ann and spoke 'her' Irish. This might have been written up in a newspaper and his bio might give us a clue as to her origin?
Did the Whites leave land records which might say 'Thomas W originally of....'
I know how hard it is to be asked to drill deeper but I know from past experience how a small remark in any one of the possibilities I have listed could make all the difference.
You mentioned Thomas brother James, did he have children/ Could there be something recorded by his line that might open this up?
Do let me know if any thing here might open up a new door for you. In the meantime I will continue to search as new records come to light.
When the Irish Civil (State) Registration records come back online we might be able to find a death for Thomas' parents.
It might be worthwhile placing a small advert in a paper in the Greely area asking for help?
Please stay in touch and I will check here when new records become available.
Warm wishes
Martine