BaFe2S3 is a quasi-one-dimensional Mott insulator that orders antiferromagnetically below 117(5) K. The application of pressure induces a transition to a metallic state, and superconductivity emerges. The evolution of the magnetic behavior on increasing pressure has up to now been either studied indirectly by means of transport measurements, or by using local magnetic probes only in the low pressure region. Here, we investigate the magnetic properties of BaFe2S3 up to 9.9 GPa by means of synchrotron Fe57 Mössbauer spectroscopy experiments, providing the first local magnetic phase diagram. The magnetic ordering temperature increases up to 185(5) K at 7.5 GPa, and is fully suppressed at 9.9 GPa. The low-temperature magnetic hyperfine field is continuously reduced from 12.9 to 10.3 T between 1.4 and 9.1 GPa, followed by a sudden drop to zero at 9.9 GPa indicating a first-order phase transition. The pressure dependence of the magnetic order in BaFe2S3 can be qualitatively explained by a combination of a bandwidth-controlled insulator-metal transition as well as a pressure enhanced exchange interaction between Fe atoms and Fe3d-S3p hybridization.