Ground arrays for ultra-high energy cosmic ray detection based on water-Cherenkov stations or scintillator modules are unavoidably limited by the saturation suffered by the counters closest to the shower axis. Reducing to a negligible level the amount of events with saturated detectors is mandatory to unambiguously record the highest energy events and to decrease the systematic uncertainties affecting the measurements. The Surface Detector Array of the upgraded Pierre Auger Observatory includes 1660 water-Cherenkov stations covered by a 3.8 m2 plastic scintillator plane. The impact point of the extensive air shower, its arrival direction and the lateral distribution of particles at the ground can be reconstructed exploiting the recorded signals and their timing. The stations will use new electronics that will process signals from both the tanks and the scintillators with increased quality. The addition of an extra small photomultiplier in each surface station and suitable photomultipliers with very large linearity in the scintillator detectors will allow us to extend the dynamic range to more than 32 times the largest signals currently measured. We describe the chosen technical solutions and discuss the expected performance of the detectors, which will be able to measure non-saturated traces as close as a couple of hundred meters to the shower core.