Polycrystalline samples of Sr1-xRExFBiS2 (RE: La, Ce, Pr, Nd, and Sm) were synthesized via the solid-state reaction and characterized using synchrotron X-ray diffraction. Although all the Sr0.5RE0.5FBiS2 samples exhibited superconductivity at transition temperatures (Tc) within the range of 2.1-2.7 K under ambient pressure, the estimated superconducting volume fraction was small. This indicated the non-bulk nature of superconductivity in these samples under ambient pressure. A dramatic evolution of the bulk superconducting phases was achieved on applying an external hydrostatic pressure. Near pressures below 1 GPa, bulk superconductivity was induced with a Tc of 2.5-2.8 K, which is termed as the low-P phase. Moreover, the high-P phase (Tc = 10.0-10.8 K) featuring bulk characteristics was observed at higher pressures. Pressure-Tc phase diagrams indicated that the critical pressure for the emergence of the high-P phase tends to increase with decreasing ionic radius of the doped RE ions. According to the high-pressure X-ray diffraction measurements of Sr0.5La0.5FBiS2, a structural phase transition from tetragonal to monoclinic also occurred at approximately 1.1 GPa. Thus, this phase transition indicates a pressure-induced superconducting-superconducting transition similar to the transition in LaO0.5F0.5BiS2. Bulk superconducting phases in Sr0.5RE0.5FBiS2 induced by the external hydrostatic pressure effect are expected to be useful for evaluating the mechanisms of superconductivity in BiCh2-based superconductors.