Enhancement of the metrological sensitivity limit through knowledge of the average energy

We consider the problem of quantum phase estimation with access to arbitrary measurements in a single suboptimal basis. The achievable sensitivity limit in this case is determined by the classical Cram'{e}r-Rao bound with respect to the fixed basis. Here we show that the sensitivity can be enhanced beyond this limit if knowledge about the energy expectation value is available. The combined information is shown to be equivalent to a direct measurement of an optimal linear combination of the basis projectors and the phase-imprinting Hamiltonian. Application to an atomic clock with oversqueezed spin states yields a sensitivity gain that scales linearly with the number of atoms. Our analysis further reveals that small modifications of the observable can have a strong impact on the sensitivity.