We present a method for finding individual excited states' energy stationary points in complete active space self-consistent field theory that is compatible with standard optimization methods and highly effective at overcoming difficulties due to root flipping and near-degeneracies. Inspired by both the maximum overlap method and recent progress in excited-state variational principles, our approach combines these ideas in order to track individual excited states throughout the orbital optimization process. In a series of tests involving root flipping, near-degeneracies, charge transfers, and double excitations, we show that this approach is more effective for state-specific optimization than either the naive selection of roots on the basis of energy ordering or a more direct generalization of the maximum overlap method. We provide evidence that this state-specific approach improves the performance of complete active space perturbation theory for vertical excitation energies. Furthermore, we find that the state-specific optimization can help avoid state-averaging-induced discontinuities on potential energy surfaces. With a simple implementation, a low cost, and compatibility with large active space methods, the approach is designed to be useful in a wide range of excited-state investigations.