He was offered the Cheshire authority job in December. He was nearing the end of his notice period when he decided to stay at Staffordshire Connects.
Crewe assistant chief executive Alan Warburton – now temporary head of comms – said recruitment for the post would continue.
Malkin said he had decided to stay at Staffordshire Connects after it won Beacon Status two weeks ago from the Office of the Deputy Prime Minister.
He joined Staffordshire Connects eight months ago. Prior to that, Malkin spent five years at Staffordshire Moorlands District Council.