The next author to have "Rio Grande" is L.A. Smith, in her very influential collection,

1888[June 1887]        Smith, Laura Alexandrine. The Music of the Waters. London: Kegan, Paul, Trench & Co.

Smith copied heavily (verbatim) from prior writing, especially from the 1869 Chambers's article and from Alden's 1882 Harper's article. Less than 1/3 of the items were collected by Smith herself.

In the case of "Rio Grande" she gives 3 versions.

One is the "fishes" version, taken directly from Alden.

The source of the other two versions is hard to pin down. For the first, one verse only, musical score is given with lyrics beneath the staves.

Another outward-bound chanty is "To Rio Grande we're bound away ;" the tune of this last-named is very mournful, as will be found in the fews bars of the melody which follows:

The ship went sailing out over the bar,
O Rio! O Rio!
They pointed her nose for the Southron Star,
And we're bound for the Rio Grande.
Then away, love, away,
Away down Rio;
Then fare you well, my pretty young girl,
We're bound for the Rio Grande.

I don't think Smith collected it in the field, because her personal collections are presented differently (as musical score WITHOUT lyrics beneath). The way she describes it as "mournful," and the following passage (replete with 'mistake' of Valparaiso/Round the Horn) are straight out of 1869 Chambers's. However, even if she had slightly changed the lyrics, that wouldn't explain where the tune came from.

This then is followed by lyrics-only to another version. It provides the "milkmaid" theme in full for the first time.

"Valparaiso," "Round the Horn," and "Santa Anna," are all much in the same style as "Rio Grande."

Solo.—"Were you ever in Rio Grande?
Chorus.—Away you Rio.
Solo.—O were you ever in Rio Grande?
Chorus.—I am bound to the Rio Grande.
      Away you Rio, away you Rio.
      Fare you well, my pretty young girl,
      I am bound to the Rio Grande.
Solo.—As I was going down Broadway Street,
Solo.—A pretty young girl I chanced to meet,
Chorus.—I am bound to Rio Grande.
Away you Rio, away you Rio,
Fare you well, my pretty young girl,
I am off to Rio Grande.
Solo.—Oh where are you going, my pretty maid?
Solo.—Oh where are you going, my pretty maid?
Chorus.—I am bound to the Rio Grande.
Away you Rio, &c.
Solo.—I am going a milking, sir, she said.
Solo.—I am going a milking, sir, she said.
Chorus.—I am bound to the Rio Grande.
Away you Rio, &c.
Solo.—What is your fortune, my pretty maid?
Solo.—What is your fortune, my pretty maid?
Chorus.—I am bound to the Rio Grande,
Away you Rio, &c.
Solo.—My face is my fortune, sir, she said.
Solo.—My face is my fortune, sir, she said.
Chorus.—I am bound to the Rio Grande.
Away you Rio, &c.
Solo.—What is your father, my pretty maid?
Solo.—What is your father, my pretty maid?
Chorus.—I am bound to the Rio Grande.
Away you Rio, away you Rio.
Then fare you well, my pretty young girl,
I am bound to the Rio Grande.
Solo.—My father's a farmer, sir, she said.
Solo.—My father's a farmer, sir, she said.
Chorus.—I am bound to the Rio Grande.
Away you Rio, &c.
Solo.—What is your mother, my pretty maid?
Solo.—What is your mother, my pretty maid?
Chorus.—I am bound to the Rio Grande.
Away you Rio, &c.
Solo.—Wife to my father, sir, she said.
Solo.—Wife to my father, sir, she said.
Chorus.—I am bound to the Rio Grande.
Away you Rio, &c.
Solo.—Then I can't marry you, niy pretty maid.
Solo.—Then I can't marry you, my pretty maid.
Chorus.—I am bound to the Rio Grande.
Away you Rio, &c.
Solo.—Nobody asked you, sir, she said.
Solo.—Nobody asked you, sir, she said.
Chorus.—I am bound to the Rio Grande.
Away you Rio, &c."

It's likely that the _Carmina Princetonia_ version (1894), upthread, ultimately derives from this.