The 90th-ranked Joint made a brisk start against Turkey's 63rd-ranked Zeynep Sonmez before succumbing 3-6 6-4 6-3.
Jones, the world No.134, also began well in her match with top seeded American Ann Li, but lost 5-7 6-0 6-3.
Australia's challenge will now solely be carried by fourth seed Talia Gibson, who meets Austrian 24-year-old Sinja Kraus in the round of 16 on Wednesday.
Also on Tuesday, American second seed Peyton Stearns bowed out when she retired injured while trailing Czechia's Katerina Siniakova 6-4 0-6 4-0.
Meanwhile, Iga Swiatek has ended her season due to a thigh injury and will miss the season-ending WTA Finals.
The six-time grand slam champion withdrew from the Ningbo Open in China having been hampered by the injury in her 7-5 6-1 loss to compatriot Maja Chwalinska in the Wuhan Open semi-finals last week.
"This has been one of the most demanding seasons of my career, with a lot of challenges, changes and lessons along the way," Swiatek said on social media.
"The past few months, or actually the past couple of years, have been incredibly intense and I feel I need to rest and recover my body fully. So that's exactly what I will do, a bit longer this time. I've decided to end my season here, which means I won't be playing any more tournaments this year."
The 25-year-old Pole, who has played at every edition of the Finals since 2021 and won the event in 2023, won one tournament this year, the Canadian Open, and failed to advance beyond the quarters at any of the four Grand Slams.
In the Ningbo Open there were victories on Tuesday for Jasmine Paolini, Diana Shnaider and Belinda Bencic.
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