Starting in the 1950s, he had his own quartet and began concentrating on a formula which gained him wider appeal for a decade. The quartet consisted of George "River Rider" Rhodes on piano, John "Broken Down" Browne on bass and Harold "Hard Nuts" Austin on drums. The most-mentioned accomplishment of this style is their version of "On The Street Where You Live", a strong-swinging treatment of the Broadway tune with a boogie-woogie jump blues feel. This effort succeeded and he began to be known to a wider audience. This led to his quartet performing on "An Evening With Fred Astaire" in 1958 and an award at the Grammy Awards of 1960, receiving the Grammy Award for Best Jazz Instrumental Album. In 1972 he made a return to more "core" jazz work with Earl Hines on the Chiaroscuro album "Back On The Street". Jones enjoyed especial popularity in France, being featured in a jazz festival in the Salle Pleyel.