  The superconducting transition of FeSe$_{1-x}$S$_x$ with three distinct
sulphur concentrations $x$ was studied under hydrostatic pressure up to
$\sim$70 kbar via bulk AC susceptibility. The pressure dependence of the
superconducting transition temperature ($T_c$) features a small dome-shaped
variation at low pressures for $x=0.04$ and $x=0.12$, followed by a more
substantial $T_c$ enhancement to a value of around 30 K at moderate pressures.
In $x=0.21$, a similar overall pressure dependence of $T_c$ is observed, except
that the small dome at low pressures is flattened. For all three
concentrations, a significant weakening of the diamagnetic shielding is
observed beyond the pressure around which the maximum $T_c$ of 30 K is reached
near the verge of pressure-induced magnetic phase. This observation points to a
strong competition between the magnetic and high-$T_c$ superconducting states
at high pressure in this system.
