We report a systematic study of longitudinal and transverse magnetoresistance in the quasi-one-dimensional organic conductor (TMTSF)(2)ClO4. We found two distinctly different quantum (rapid) oscillations (RO's) at different temperature regions in the field-induced spin-density-wave (FISDW) phase. For temperature sweeps in a fixed held, a resistance peak is observed at T* which depends on the field. Surprisingly, the type of RO is correlated with T*(H). The difference between two oscillations leads us to establish subphases in the main FISDW phase of (TMTSF)(2)ClO4. We also found the low-field boundary of McKernan's 3.5 K phase from a sudden increase of resistance. Our results conclude that the separate SDW transition on each pair of the Fermi surfaces is possibly responsible for subphases in the FISDW phase.