Irlam, 26, made 113 appearances during a five-year spell at the Cheshire outfit, scoring 13 goals.
He has also featured for Runcorn Town, where he played over 100 games, Ashton Town, Barnton and City of Liverpool.
On his move to Celtic, Irlam said: "I’m happy and end excited to join Stalybridge Celtic and looking forward to pushing on this season and hopefully getting the club back up there.
"The manager and players have welcomed me with open arms. It’s now time to crack on and win football matches."
Stalybridge manager James Kinsey is delighted with his new acquisition and is excited to see him in action.
"Jack is somebody we have watched for a long time now. He is really strong on the ball and can carry the ball well," Kinsey said.
"He has experience in this league and ticks the boxes for what we want in the dressing room.
"He has the ability to drag us through games and we’re excited to see him push on here."
Photo credit: Xenia Simpson Photography