  BaFe$_2$S$_3$ 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 BaFe$_2$S$_3$ up to 9.9\,GPa by means of synchrotron $^{57}$Fe
M\"ossbauer 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
BaFe$_2$S$_3$ 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 Fe 3\textit{d}-S 3\textit{p}
hybridization.
