We present a new analysis for estimating the depth of the maximum of air-shower profiles, $X_\mathrm{max}$, to investigate the evolution of the ultra-high-energy cosmic ray mass composition from 3 to 100 EeV. We use a recently developed deep-learning-based technique for the reconstruction of $X_\mathrm{max}$ from the data of the surface detector of the Pierre Auger Observatory. To avoid systematic uncertainties arising from hadronic interaction models in the simulation of surface detector data, we calibrate the new reconstruction technique with observations of the fluorescence detector. Using the novel analysis, we have a 10-fold increase of statistics at $E>5$ EeV with respect to fluorescence detector data. We are able, for the first time, to study the evolution of the mean and standard deviation of the $X_\mathrm{max}$ distributions up to 100 EeV. We find an excellent agreement with fluorescence observations and confirm the increase of the mean logarithmic mass <lnA> and a decrease of the $X_\mathrm{max}$ fluctuations with energy. The $X_\mathrm{max}$ measurement at the highest --- so far inaccessible --- energies is consistent with a pure mass composition and a mean logarithmic mass of around $\sim3$ (estimated using the Sibyll 2.3d and the EPOS-LHC hadronic interaction models). Furthermore, with the increase in statistics, we find indications for a structure beyond a constant elongation rate in the evolution of $X_\mathrm{max}$.