The detection of ultra-high-energy cosmic rays by means of giant detector arrays is often limited by the saturation of the recorded signals near the impact point of the shower core at the ground, where the particle density dramatically increases. The saturation affects in particular the highest energy events, worsening the systematic uncertainties in the reconstruction of the shower characteristics. The upgrade of the Pierre Auger Observatory, called AugerPrime, includes the installation of an 1-inch Small PhotoMultiplier Tube (SPMT) inside each water-Cherenkov station (WCD) of the surface detector array. The SPMT allows an unambiguous measurement of signals down to about 250m from the shower core, thus reducing the number of events featuring a saturated station to a negligible level. In addition, a 3.8m$^2$ plastic scintillator (Scintillator Surface Detector, SSD) is installed on top of each WCD. The SSD is designed to match the WCD (with SPMT) dynamic range, providing a complementary measurement of the shower components up to the highest energies. In this work, the design and performances of the upgraded AugerPrime surface-detector stations in the extended dynamic range are described, highlighting the accuracy of the measurements. A first analysis employing the unsaturated signals in the event reconstruction is also presented.