The Emmy and Golden Globe-winning star is also known for his role in the film Manhunter and his stage work with the Royal Shakespeare Company (RSC).
He has been performing in Make it Happen, the story of the rise and fall of the Royal Bank of Scotland (RBS), in Edinburgh.
On postponing his tour, Cox said: "I just thought 'I'll do it some time', [October is] just too soon."
He said he is due to take his new film Glenrothan - the first he has directed - to the Toronto Film Festival next month.
"It's been very busy and I just thought 'I can't be doing it'," he said.
The tour would have also seen Cox perform at the Corn Exchange in Ipswich on 3 October and the Cliffs Pavilion in Southend-on-Sea on 20 October.