The Trio appeared at the first Newport Folk Festival in 1959. Whether there is a video of that concert I don't know. There is an album, "The Kingston Trio Live at Newport"
The Trio did appear at the well established Newport Jazz Festival in April of 1959, as Bill Bush notes in his Greenback Dollars book, as well as on the cover of premier jazz magazine Downbeat in July of 1959. The connection between the KT and jazz may seem remote to us now, but both serious jazz and folk were considered outliers from the pop mainstream at the time.
It's also worth noting that the liner notes for At Large were written by Nat Hentoff, as serious a student and proponent of jazz as U.S. journalism has ever had.
Worth noting as well is that the success at the "folk afternoon" at the spring '59 Newport jazz fest (the KT plus Odetta) was the prime reason that promoter and Newport jazz founder George Wein went ahead with his long-planned Newport Folk Festival later in 1959. He was sure that he needed both the Trio and Odetta at the late summer folk festival to make it a financial success, and both were headliners (with Pete Seeger and others).
We may tend to forget that getting high quality reproduceable video in 1959 was very expensive, requiring essentially a motion picture camera that had a sound strip on the borders of the film so that audio and visuals would be coordinated.
I seem to recall that a few individual numbers were filmed from both 1959 Newport Festivals by a number of the acts, including the KT, but it'd take a bit more time than I have right now to run that down.