We report the most precise measurements to date of the strong-phase parameters between D0 and D¯0 decays to KS,L0π+π-using a sample of 2.93 fb-1 of e+e- A nnihilation data collected at a center-of-mass energy of 3.773 GeV with the BESIII detector at the BEPCII collider. Our results provide the key inputs for a binned model-independent determination of the Cabibbo-Kobayashi-Maskawa angle γ/φ3 with B decays. Using our results, the decay model sensitivity to the γ/φ3 measurement is expected to be between 0.7° and 1.2°, approximately a factor of three smaller than that achievable with previous measurements, based on the studies of the simulated data. The improved precision of this work ensures that measurements of γ/φ3 will not be limited by knowledge of strong phases for the next decade. Furthermore, our results provide critical input for other flavor-physics investigations, including charm mixing, other measurements of CP violation, and the measurement of strong-phase parameters for other D-decay modes.