The pressure dependence of the structural, magnetic, and superconducting transitions and of the superconducting upper critical field were studied in sulfur-substituted Fe(Se1−xSx). Resistance measurements were performed on single crystals with three substitution levels (x=0.043, 0.096, 0.12) under hydrostatic pressures up to 1.8 GPa and in magnetic fields up to 9 T and were compared to data on pure FeSe. Our results illustrate the effects of chemical and physical pressure on Fe(Se1−xSx). On increasing sulfur content, magnetic order in the low-pressure range is strongly suppressed to a small domelike region in the phase diagrams. However, Ts is much less suppressed by sulfur substitution, and Tc of Fe(Se1−xSx) exhibits similar nonmonotonic pressure dependence with a local maximum and a local minimum present in the low-pressure range for all x. The local maximum in Tc coincides with the emergence of the magnetic order above Tc. At this pressure the slope of the upper critical field decreases abruptly, which may indicate a Fermi-surface reconstruction. The minimum of Tc correlates with a broad maximum of the upper critical field slope normalized by Tc.