That final verse is present in most broadside copies, and seems likely to have been part of the song from the start. Rinordine is in all editions of the Forget Me Not Songster except for a variant of (probably) the 1850s, but no tune is specified. Nafis & Cornish editions are likely between 1843 and 1851, but the whole publishing history is quite complicated.

See Norm Cohen's 'The Forget-Me-Not Songsters and Their Role in the American Folksong Tradition' in American Music vol 23 no 2, Summer 2005, Society of American Music & University of Illinois Press. It is the standard work on the subject, and a mine of information.