...Pursuing it with weary feet,
Until it joins some larger way,
Where many paths and errands meet.
And whither then? I cannot say.

Since Errantry is not in LotR and nor is Progress in Bimble Town (another favourite).

I'll have to change my answer to the wonderful poem 'Where There's a Whip There's a Way' from the Rankin/Bass Return of the King.

Was it 'Errantry' that Tolkien published somewhere obscure and then later found someone came to him with a mildly garbled version of the poem hoping (in all innocence) that he might know who the author was?

As for 'Errantry': it is a most odd coincidence that you should ask about that. For only a few weeks ago I had a letter from a lady unknown to me making a similar enquiry. She said that a friend had recently written out for her from memory some verses that had so taken her fancy that she was determined to discover their origin. He had picked them up from his son-in-law who had learned them in Washington D.C. (!); but nothing was known about their source save a vague idea that they were connected with English universities. Being a determined person she apparently applied to various Vice-Chancellors, and Bowra [then vice-chancellor of Oxford] directed her to my door. I must say that I was interested in becoming 'folk-lore'. Also it was intriguing to get an oral version - which bore out my views on oral tradition (at any rate in early stages): sc. that the 'hard words' are well preserved, and the more common words altered, but the metre is often disturbed.

233 (I: 246). Eärendil was a mariner... - The long and complex history of this poem is discussed at length by Christopher Tolkien in The Treason of Isengard, Chapter 5. Here it will suffice to say that probably in the early 1930s Tolkien wrote a set of verses called Errantry. The earliest extant version begins: There was a merry passenger, / a messenger, an errander.
This developed through five further texts until the work was published in the Oxford Magazine for 9 November 1933 (reprinted in The Treason of Isengard). When, years later, Tolkien wanted a poem for Bilbo to recite at Rivendell in The Lord of the Rings, he developed Errantry further through fifteen more texts, until it was concerned with Eärendil, a pivotal figure in his mythology. The earlier of these versions begin 'There was a merry messenger' (or a variant); the six later texts, probably after a long interval of time, begin 'Eärendel [or Eärendil] was a mariner'. The third version in this second category was published in The Lord of the Rings in 1954; but three more, further revised versions followed. Christopher Tolkien believes that these final texts were mislaid by his father, and not to be found when the typescript of The Lord of the Rings had to be sent to the publisher, so that a less developed version came into print, 'as it should not have done.
It looks also as if these lost texts did not turn up again until many years had passed, by which time my father no longer remembered the history The Treason of Isengard, p. 103).
The Lord of the Rings: A Reader’s Companion, by Wayne G. Hammond and Christina Scull. London: HarperCollins; Boston: Houghton Mifflin, 2005. Revised edition, London: HarperCollins, 2008. Third edition, London: HarperCollins, 2014.