Manager Brian Laws admitted the arrival of Andy Dawson from Hull means he is unable to offer Newey, 30, a new deal.
"There is always a downside," Laws told the club website.
"Tom is a terrific lad but unfortunately we couldn't offer him a deal.
"We want to wish him well. He was fantastic for us last year and did everything possible to try and earn a contract.
"If our finances were in a better shape then I'd be doing that anyway."
Laws, who has been linked with a move for Watford striker Chris Iwelumo, is confident of bringing in another new signing in the next few days.
"We're hoping to conclude one in the early part of next week, which is encouraging," he added.
"This is about bringing players who we feel will embrace what we want to do.
"We want all the right ingredients to form a good team together. Experience, craft and guile is key."
- Sports & Recreation
- Andy Dawson