We present a de Haas--van Alphen quantum oscillation study of the Dirac nodal-line semimetal HfSiS up to 32 T to unravel the structure of the high-frequency magnetic breakdown spectrum that was previously obscured in transport experiments. Despite a threefold enhanced gap between adjacent electron and hole pockets relative to the sister compound ZrSiS, a large number of large-area magnetic breakdown orbits enclosing the nodal-loop are identified. All breakdown orbits are assigned by extracting their cyclotron masses. Moreover, one additional low-frequency magnetic breakdown orbit, previously absent in ZrSiS, is observed and attributed to the larger spin-orbit interaction in HfSiS.