  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.
