We investigate several equivalent notions of the Jost solution associated with a unitary
CMV matrix and provide a necessary and sufficient conditions for the Jost solution to
consist of entire functions of finite growth order in terms of super exponential decay of
Verblunsky coefficients. We also establish several one-to-one correspondences between CMV
matrices with super-exponentially decaying Verblunsky coefficients and spectral data
associated with the first component of the Jost solution.