The tune was written in 1942. The up-a-minor-third tag doesn't appear in any early versions that I could find (Jimmy Dorsey, Harry James). And although I don't have the original sheet music, I did check the chart in the Hal Leonard Real Jazz Standards Fake Book, which shows original sheet-music changes, and the tag didn't occur there.
Browsing Youtube, the earliest usage I could find was in this Dave Brubeck version, recorded 12/14/53. He uses it as both intro and tag (or "outro," as we sometimes say):
Interestingly, Horace Silver uses a similar intro/outro approach in this version, recorded 11/23/53. Horace doesn't use the melody line, but does use the up-a-minor-third device in both intro and outro:
Wikipedia lists this track as having been released in 1953, but does not give an exact date. It's possible, but not too likely, that Brubeck had heard Silver's version prior to his own recording of "I Remember You." My guess is that neither Brubeck nor Silver actually originated this tag, but that it was "in the air" by late 1953. If any readers know of an earlier version using this tag, please let me know in the comments.
While I was listening to these recordings, my wife recalled that there had been a country-sounding version on Los Angeles pop radio in the 1960s. She mght have been remembering one of these versions - Frank Ifield (#1 on the charts in England in 1962), or Slim Whitman, who covered Ifield's version in 1966. Both versions use the up-a-minor-third tag.
Here are the Beatles in an early recording in Hamburg, covering Ifield's version, and again with the tag: