Frederick Montague, M.P., created the precursor to the technique—which used no cover card—and published it as “The 'Bluff' Pass and Palm” in Westminster Wizardry, 1928, p. 75. (Montague followed the control with palming off the selection.)
The sleight, employing a cover card, was quietly circulating, but didn't reach print until 1936. For example, Jean Hugard recorded it in notes dated April 1932, giving no attribution, but recognizing the similarity of the sleight to Montague's.
Four years after Hugard made his notes on the sleight, Tommy Tucker published a description of it in his booklet What Next!, 1936, p. 24. Tucker included the dribble finesse in his description.