Quantum Cramer-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.