In this work we construct a class of coanalytic Toeplitz operators, which have an infinitedimensional closed subspace, where any non-zero vector is hypercyclic. Namely, if for a function ϕ which is analytic in the open unit disc D and continuous in its closure the conditions ϕ(T)∩T ≠ ∅ and ϕ(D)∩T ≠ 0 are satisfied, then the operator ϕ(S*) (where S* is the backward shift operator in the Hardy space) has the required property. The proof is based on an application of a theorem by Gonzalez, Leon-Saavedra and Montes-Rodriguez.