Ellen Tapply appears as Mrs. Tapply along with Annie and Daisy on the manifest for the steamship Bolivia in June 1881. The Bolivia was on the Anchor line and this journey took them in steerage from London to Castle Garden Immigration Station in New York. (Ellis Island wasn't opened yet) The two pictures above are of sister ships on that line. I'm thinking the one on the left is the closest image.
I've tried using just his age, just his first name, just his last name, and variations on that. I also tried using Ellen's maiden name and her mother's surname. I figure at the very least he was traveling under an alias.
I also found the documents for his father James Henry Tapply and his brother-in-law Stephen Hodge. Nothing so far for Thomas. In those days wives and children were grandfathered in when the man took his citizenship. That's a shame, because there's always interesting information.
At least with this latest information we know that he didn't sail back and forth. It settles the question of the timing of Daisy's birth. (when I thought he has arrived earlier, I wondered about that) I still think about that young woman traveling with a tiny infant and a 3-year-old in steerage for a long ocean voyage. The motivation for a new start must have been strong. This doesn't completely discount the story Charles told his children and there is still a bit of mystery there for me to pursue.
Good. I like a good mystery.