Pinnington Jones claimed his first match win at a Grand Slam with victory over Tomas Martin Etcheverry at Wimbledon last year.
A close friend of British number one Jack Draper, Pinnington Jones completed his junior season at Texas Christian University in May but opted against returning for a final year in order to pursue a full-time tennis career.
Elsewhere, Elisabetta Cocciaretto progressed to the last 16 of a WTA 1000 event for the first time with a dominant win over Coco Gauff at the Qatar Open.
Italian Cocciaretto saved four of the five break points she faced in a 6-4 6-2 victory over French Open champion Gauff.
Elena Rybakina won her first match since her Australian Open triumph with a 6-2 6-4 victory over China’s Wang Xinyu, while world number two Iga Swiatek breezed past Janice Tjen 6-0 6-3.
Britain’s Emma Raducanu retired from her first-round match against Camila Osorio earlier in the week.