A New York Times article next mentions "Rio Grande," and mentions the "milkmaid" theme, I believe, for the first time in connection with this chanty.

1884[Jan]        Unknown. "Minstrelsy on the Sea." The New York Times, 27 (Jan. 1884). pg. 10.

It says,

The song in which a young man meets a pretty maid, who, upon being cross-examined, informs him that her face is her fortune, and in a very pert and forward manner says: "Nobody asked you, Sir" when he announces his disinterested Intention of marrying her, has, after some alterations and renovations been transformed into a shanty, with the following somewhat irrelevant chorus: "I was bound for the Rio Grande."