The potential impact of optimization can be hindered by an ill-conditioned problem formulation that leads to slow convergence or even convergence failure. The goal of this work is to address ill-conditioning indirectly, without creating a new optimization library or modifying an existing library. Our non-intrusive approach creates a transformation that maps the user's variables to working variables outside of the optimization library, such that the problem in the transformed space is better conditioned. The transformation is constructed using a matrix-free Arnoldi's method that estimates the spectrum of the Hessian corresponding to the largest magnitude eigenvalues. Numerical experiments demonstrate that, for ill-conditioned problems, the Hessian-based transformation can improve run times by an order of magnitude or more, and it can successfully converge some problems that fail without the transformation.