Operators on function spaces acting by composition to the right with a fixed self-map ϕ are called composition operators. We denote them Cϕ. Given ϕ, a hyperbolic disc automorphism, the composition operator Cϕ on the Hilbert Hardy space H2 is considered. The bilateral cyclic invariant subspaces Kf, f ∈ H2, of Cϕ are studied, given their connection with the invariant subspace problem, which is still open for Hilbert space operators. We prove that nonconstant inner functions u induce non–minimal cyclic subspaces Ku if they have unimodular, orbital, cluster points. Other results about Ku when u is inner are obtained. If f ∈ H2 \ {0} has a bilateral orbit under Cϕ, with Cesàro means satisfying certain boundedness conditions, we prove Kf is non–minimal invariant under Cϕ. Other results proving the non–minimality of invariant subspaces of Cϕ of type Kf when f is not an inner function are obtained as well.