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The simulation of excited states at low computational cost remains an open challenge for electronic structure (ES) methods. While much attention has been given to orthogonal ES methods, relatively little work has been done to develop nonorthogonal ES methods for excited states, particularly those involving nonorthogonal orbital optimization. We present here a numerically stable formulation of the Resonating Hartree–Fock (ResHF) method that uses the matrix adjugate to remove numerical instabilities arising from nearly orthogonal orbitals, and as a result, we demonstrate improvements to ResHF wavefunction optimization. We then benchmark the performance of ResHF against complete active space self-consistent field in the avoided crossing of LiF, the torsional rotation of ethene, and the singlet–triplet energy gaps of a selection of small molecules. ResHF is a promising excited state method because it incorporates the orbital relaxation of state-specific methods, while retaining the correct state crossings of state-averaged approaches. Our open-source ResHF implementation, yucca, is available on GitLab.