The 25-year-old arrives at Bower Fold having made 47 appearances for his former club since October 2022.
The experienced defender is a two-time promotion winner with Warrington Rylands, having won promotion from the NWCFL Premier Division and Pitching In NPL West Division.
He was also part of Rylands' FA Vase-winning squad.
Starting his career at Oldham Athletic, Tinning has also featured for FC United of Manchester, Abbey Hey, 1874 Northwich and Witton Albion, before spending the last 12 months with Bury.
“I’m looking forward to getting started at Stalybridge Celtic," he said following the move.
"Kinsey and Dan have put a great squad together and hopefully I can bring some more quality to the team.”
Manager James Kinsey said: “Jack will provide even more improvement to our defensive unit.
"He brings left-footed balance and can play at both centre back or left-back. He is a local lad too, which is important to the club.
"He had a few options, so we’re delighted that he has chosen to come here.”