The next "Rio Grande" I have come in

1871        Adams, Nehemiah. _A Voyage Around the World_. Boston: Henry Hoyt.

The author was the father of RC Adams, a noted writer about chanties in his famous travelogue _On Board the Rocket_.

The song is mentioned here on the American ship Golden Fleece on a voyage, circa late 1869-70, from Boston to San Francisco, Hong Kong, and the Philippines. It was allegedly used for pumping. Only a phrase is mentioned.

...the boatswain's "Pumpship " at evening, when twelve or fifteen men entertain you with a song. Every tune at the pumps must have a chorus. The sentiment in the song is the least important feature of it, — the celebration of some portion of the earth or seas, other than here and now: "I wish I was in Mobile Bay," "I'm bound for the Rio Grande," with the astounding chorus from twenty-eight men, part of whom the fine moonlight and the song tempt from their bunks, is an antidote to monotony.

Next,

1876(Sept.) Symondson, F.W.H. _Two Years Abaft the Mast_. Edinburgh and London: William Blackwood and Sons.

This one attributes "Rio Grande" during a voyage circa 1874 in the English ship Sea Queen, to Sydney. It was sung at the anchor capstan.

Whilst heaving up anchor prior to the tug towing us to the wharf, we had some good "chanties"—for Jack's spirits are at their highest at the thoughts of a run ashore. The "chanty" known under the name of " The Rio Grande" is particularly pretty, the chorus being:—

"Heave away, my bonny boys, we are all bound to Rio.
          Ho! and heigho!
Come fare ye well, my pretty young girl,
      For we're bound to the Rio Grande."