Let \(\mathcal{H}\) be an infinite-dimensional separable Hilbert space. For \(T\) a bounded linear operator on \(\mathcal{H}\), recall that a closed subspace \(W\subset\mathcal{H}\) is a hyperinvariant subspace of \(T\) if \(W\) is invariant under any operator commuting with \(T\). We denote by Hlat\((T)\) the hyperinvariant subspace lattice of \(T\). In the paper under review, the authors consider the class \(\mathcal{L}_0(\mathcal{H})\) of cyclic quasianalytic contractions, and the subclass \(\mathcal{L}_{1}(\mathcal{H})\subset\mathcal{L}_0(\mathcal{H})\) containing operators whose quasianalytic spectral sets are the unit circle. It is known by the work of the first author [J. Funct. Anal. 246, No. 2, 281--301 (2007; Zbl 1123.47008)] that every operator in \(\mathcal{L}_{1}(\mathcal{H})\) has a rich invariant subspace lattice. The main result of the present paper asserts that for every operator \(T\in\mathcal{L}_0(\mathcal{H})\), there exists an operator \(T_1\in\mathcal{L}_1(\mathcal{H})\) commuting with \(T\). It then follows that the identity Hlat\((T) =\) Hlat\((T_1)\) holds. As a consequence, the Hyperinvariant Subspace Problem (HSP) in the class \(\mathcal{L}_0(\mathcal{H})\) is equivalent to the HSP in the class \(\mathcal{L}_1(\mathcal{H})\). The operator \(T_1\) in the main theorem is given by \(T_1=f(T)\), where \(f\) is an appropriate \(H^{\infty}\)-function on the unit disk. The existence of such an \(f\) is proved by using tools from potential theory.