Jones had been pushing for a return to the starting line-up in Tuesday's Carling Cup clash at home to Burnley after recovering from a groin injury to make a substitute appearance at Priestfield.
However, he will now visit a specialist to determine the extent of the damage to his hand.
Director of sport Chris Turner told the club's official website: "He is going to see a specialist to see how serious the injury is but it's likely he'll be out for around three weeks.
"We need to see whether he can have a cast put on to enable him to play or not. Gary Liddle had a similar injury a few seasons ago and although he couldn't play for the first few weeks, he could then play with a cast on, so we'll look at that option."
In contrast, Turner has been boosted by the return to training of defender Ben Clark and forward Colin Larkin after hernia and thigh problems respectively.