The 22-year-old joined the Eagles from Huddersfield Town on the penultimate day of the summer transfer window, but looks poised for a lengthy spell on the sidelines after suffering an injury in training.
Hunt told reporters: "Absolutely devastated to have broken my ankle in training. Got to take the positives, clean break and no op needed.
"I'll do everything I can to be back playing as soon as possible. Massive thanks to everyone at CPFC. Chairman, manager, coaching staff, physios and the lads have been unreal."
The versatile Yorkshireman, who can play at either full-back or on the right side of midfield, made 121 appearances for Huddersfield, who gained £2m from Hunt's move to Selhurst Park.