In this report, we experiment a variant of the two-level ideal algorithm for parametric shape optimization that was proposed in \cite{Desideri6}. In the linear case, the method, referred to as the $Z'$ method, employs a permutation operator to rearrange the eigenstructure in such a way that the new high-frequency modes are associated with large eigenvalues. As a result, the classical steepest-descent iteration can be viewed as a Jacobi-type smoother, and standard multilevel strategies be applied. An alternate method is also tested based on odd-even decoupling ($L'$ method). For a linear model problem, both new methods are found efficient and superior to the original formulation, but the $Z'$ method is more robust. Similar numerical results are obtained for a nonlinear model problem by considering the eigensystem of the Jacobian matrix.