Here's a passage from Chris Albertson's superb liner notes that accompany the Mosaic box, The Complete Clef/Verve Count Basie Fifties Studio Recordings:
"Organist Wild Bill Davis [pictured] wrote this arrangement, giving it a decided organ feel and making other departures that its composer surely welcomed... The touch that makes this arrangement such a hit with live audiences is the repetitive "one more time" ending, which came about during a joint Birdland appearance by the band and Davis' organ trio.
"When the Basie band ended the tune, Davis turned to Basie and shouted, "One more time." The band complied, and the audience loved it. The routine became embedded in the chart, often going on for several "one more time" requests.
"The original intent was to to have the band record the tune with Wild Bill's trio, recreating the interaction that grew out of the Birdland gig. But when something came up to prevent Wild Bill from making the date, Basie decided to go it alone. He would later refer to that as a lucky break, for April in Paris became the band's biggest selling instrumental."
Here's Wild Bill Davis being introduced by Duke Ellington and playing his arrangement of April in Paris: