Thomas Morgan began playing the cello at 7, eventually switching to upright-bass at 14. In 2003 he received his bachelor's degree in Music from the Manhattan School of Music, where he studied with Harvie Swartz and Garry Diall. He has also studied briefly with Ray Brown and Peter Herbert. Morgan has worked with David Binney, Steve Coleman, Joey Baron, Josh Roseman, Brad Shepik, Steve Cardenas, Timuçin Şahin, Kenny Wollesen, Gerald Cleaver, Adam Rogers and Kenny Werner. He has also collaborated with Jakob Bro, Dan Tepfer, Jim Black, John Abercrombie, and Masabumi Kikuchi, and he has performed with the Sylvie Courvoisier-Mark Feldman Quartet.
Morgan was featured prominently on the 2017 ECM album Small Town in a duet setting with guitarist Bill Frisell. The album documents a 2016 live performance at the Village Vanguard.
In 2014, Morgan's own trio, featuring keyboardist Pete Rende and drummer Dan Weiss, was reviewed in The New York Times by jazz critic Ben Ratliff, who called Morgan, "a jazz musician whose presence in a recording or a performance almost automatically [makes] it worth your time."