Three spare modules of the ATLAS Tile Calorimeter were exposed to test beams from the Super Proton Synchrotron accelerator at CERN in 2017. The detector’s measurements of the energy response and resolution to positive pions and kaons, and protons with energies ranging from 16 to 30 GeV are reported. The results have uncertainties of a few percent. They were compared to the predictions of the Geant4-based simulation program used in ATLAS to estimate the response of the detector to proton-proton events at the Large Hadron Collider. The determinations obtained using experimental and simulated data agree within the uncertainties.