Interlayer resistance is reported in a layered organic superconductor β″-(BEDT-TTF)2SF5CH2CF2SO3 with Tc≈5 K. At 0.7 K in parallel magnetic fields, the interlayer resistance has nonzero values in a wide field region from 2 T to the upper critical field (∼13T), which can be ascribed to the Josephson vortex (JV) dynamics. Significant hysteresis between 2 T and 8 T and a steplike structure between 4 T and 8 T are found, both of which become less evident with increasing temperature. In the second field-derivative curves, successive small dips are observed between 10 T and 12 T, which are ascribed to the commensurability (CM) effect between the JV lattice and the wavelength λFFLO of the order parameter oscillation in the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase. The CM effect is observed only in nearly parallel fields, showing that the FFLO phase is stabilized only in a limited field angle region. The temperature-field phase diagram and λFFLO values are obtained.