Jones was a second-half substitute in Saturday's 1-0 win at Gillingham after recovering from a groin problem and showed no ill-effects.
However, he has since been diagnosed with a broken hand and has been ruled out for at least three weeks.
Director of sport Chris Turner named a full-strength line-up in the first round and was rewarded with a 1-0 extra-time victory at Coventry, and he will be targeting another notable scalp on Tuesday night.
Pools knocked West Brom out of the competition at the same stage last season, while Stoke were beaten 2-0 at Victoria Park in the FA Cup, so the League One side will not be short of confidence.
With Jones injured, James Brown is unlikely to now be rested despite starting two games last week.
Turner has no other fresh injury or suspension concerns.