  Quantum Cram\'er-Rao bound is the ultimate limit of the mean squared error
for unbiased estimation of an unknown parameter embedded in a quantum state.
While it can be achieved asymptotically for large number of quantum state
copies, the measurement required often depends on the true value of the
parameter of interest. This paradox was addressed by Hayashi and Matsumoto
using a two-stage approach in 2005. Unfortunately, their analysis imposes
conditions that severely restrict the class of classical estimators applied to
the quantum measurement outcomes, hindering applications of this method. We
relax these conditions to substantially broaden the class of usable estimators
at the cost of slightly weakening the asymptotic properties of the two-stage
method. We apply our results to obtain the asymptotics of quantum-enhanced
transmittance sensing.
