Joining the band on tour are Beddy Rays, who will open all shows, with Teenage Joans appearing in Sydney and Brisbane and Secret World set for the Melbourne date.
The April 27 episode -- which will feature guest judge Nikki Glaser -- will be the first night entirely dedicated to Swift's music in the show's 24 seasons.
Cook, who served in the role for nearly 15 years, will remain involved with the company as executive chairman. John Ternus is set to succeed him as CEO.