  Critical metrology relies on the precise preparation of a system in its
ground state near a quantum phase transition point where quantum correlations
get very strong. Typically this increases the quantum Fisher information with
respect to changes in system parameters and thus improves the optimally
possible measurement precision limited by the Cram\'er-Rao bound. Hence
critical metrology involves encoding information about the unknown parameter in
changes of the system's ground state. Conversely, in conventional metrology
methods like Ramsey interferometry, the eigenstates of the system remain
unchanged, and information about the unknown parameter is encoded in the
relative phases that excited system states accumulate during their time
evolution. Here we introduce an approach combining these two methodologies into
a unified protocol applicable to closed and driven-dissipative systems. We show
that the quantum Fisher information in this case exhibits an additional
interference term originating from the interplay between eigenstate and
relative phase changes. We provide analytical expressions for the quantum and
classical Fisher information in such a setup, elucidating as well a
straightforward measurement approach that nearly attains the maximum precision
permissible under the Cram\'er-Rao bound. We showcase these results by focusing
on the squeezing Hamiltonian, which characterizes the thermodynamic limit of
Dicke and Lipkin-Meshkov-Glick Hamiltonians.
