The longitudinal magnetoresistance of the unusual Bechgaard salt, (TMTSF)₂FSO₃, has been studied up to 33 T under various pressures. In this compound, a single series of the Shubnikov–de Haas (SdH) oscillation is very pronounced for pressures between 5.2 and 11.8 kbar where the zero-field ground state is superconducting. Unlike the case of the rapid oscillations in most Bechgaard salts, the temperature and magnetic field dependence of the oscillations is in good agreement with the Lifshitz–Kosevich formula, implying the two-dimensional closed orbital motion of the electrons. The effective mass of the electrons and Dingle temperature are estimated as 1.4 ± 0.05 m₀ and 1.6–2.4 K, respectively. For an origin of the closed orbits, it is suggested that incomplete nesting between open Fermi surfaces is induced by pressure. Discontinuous change of the frequency, amplitude, and Dingle temperature of the oscillations around 9 kbar indicates that the electronic state below and above this pressure is different. According to the features of the SdH oscillations along with the phase diagram of (TMTSF)₂FSO₃, the pressure dependence is divided into three regions.