Hird, 25, was signed by ex-Spireites manager John Sheridan last summer on a two-year contract after leaving Doncaster.
He was captain and was a regular at the heart of the defence, although he has also featured at right-back and in midfield.
However, Hird was left out of the side by new Blues boss Cook towards the end of the season.
He made 44 appearances for the League Two club and scored two goals.
The news comes the day after seven players were told they will be allowed to leave the Proact Stadium this summer as Cook wastes no time in reshaping his squad following the disappointment of failing to mount a promotion challenge.