In Bill Kalush's essay, “Sleight of Hand With Playing Cards Prior to Scot's Discoverie” in Puzzler's Tribute, 2002, p. 119, he describes a Monte-like scam being played in 1408. It used more than three cards, but the general outlying gameplay and underlying deception were the same. No mechanics were detailed at the time, so there's no way of knowing if they used the Hype toss for the switch, or a different technique.
The earliest description of a Hype move wasn't with single cards, but rather with packets (which was later reinvented by Paul Harris as Flip Flop Plop; see Jerry Mentzer's The Magic of Paul Harris, 1976, p. 21). It was described in the anonymous Asti Manuscript, c. 1700, p. 63 of the Pieper translation. This manuscript was translated in Gibecière, Vol.8 No.1, Winter 2013, p. 29-234.
The technique with single cards in a Monte context was described in Jean Eugène Robert-Houdin's Les Trickeries des Grecs Dévoilées, 1861, p. 41 of the Hoffmann translation. It was later described with the now-current grip in Professor Hoffmann's Modern Magic, 1876, p. 61.
The game of Three-Card Monte itself probably dates prior to 1795. According to M. P. Adams in The Rich Uncle from Fiji, 1911, p. 40, it was in this year that the Monte hustlers were cleared out of British racecourses, suggesting that the game was prevalent at the time. A four-card scam predates the use of the Hype move, and was detailed in Henri Decremps's Testament de Jerome Sharp, 1785, p. 179-80 of the Hugard translation. In it, the four cards are arranged in a square formation and then zig-zagged around to each of the four positions as if to lose the player in confusion. Of course, it is a scam because the money card is never put on the table in the first place; it is switched for an indifferent card as it's removed from the pack, using a glide.