Sensing a classical signal using a linear quantum device is a pervasive application of quantum-enhanced measurement. The fundamental precision limits of linear waveform estimation, however, are not fully understood. In certain cases, there is an unexplained gap between the known waveform-estimation Quantum Cram'er-Rao Bound and the optimal sensitivity from quadrature measurement of the outgoing mode from the device. We resolve this gap by establishing the fundamental precision limit, the waveform-estimation Holevo Cram'er-Rao Bound, and how to achieve it using a nonstationary measurement. We apply our results to detuned gravitational-wave interferometry to accelerate the search for post-merger remnants from binary neutron-star mergers. If we have an unequal weighting between estimating the signal’s power and phase, then we propose how to further improve the signal-to-noise ratio by a factor of (\sqrt2) using this nonstationary measurement.