The 29-year-old former Bolton right-back has made three appearances for Stanley to date but suffered severe concussion during last weekend's goalless draw with Plymouth and is expected to miss Saturday's trip to York.
"I'm very pleased Nicky's staying," said Reds manager Leam Richardson in a club statement. "He's a very good pro and a very good player and offers a lot of experience which we need.
"Before he got injured on Saturday he was playing very well and right-back is a position we've looked to fill for a lot of the season.
"To tie somebody of Nicky's experience and quality down is fantastic news for us."
And Richardson is pleased by Hunt's progress in the early stages of his recovery.
"Nicky's improving quite fast which is pleasing for us and hopefully he'll be back in contention sooner rather than later.
"With the severity of the injury you've got to leave it to the player and the medical team. I've known Nicky for a number of years now and as soon as he's ready to play he'll tell us he's ready to play."