Before Louis Cato took over for his friend Jon Batiste as bandleader of “The Late Show with Stephen Colbert” in 2022, he put out records that swerved from funk to jazz to blues and soul, with some R&B and Latin influences along the way.

Mr. Cato, 38, is still making his own music when he’s not onstage revving up the “Late Show” crowd. Most Sunday afternoons after he and his 12-year-old daughter, Gianna, part ways — he and Gianna’s mother share custody — he’s hunkered in his home studio in Midtown recording “Cato covers,” his version of a beloved song, which he shares with his Instagram followers each Monday.