  We present magnetization and muon-spin spectroscopy measurements of N\'{e}el
skyrmion-host GaV$_4$S$_8$ under the application of hydrostatic pressures up to
$P=2.29$ GPa. Our results suggest that the magnetic phase diagram is altered
with pressure via a reduction in the crossover temperature from the cycloidal
(C) to ferromagnetic-like state with increasing $P$, such that, by 2.29 GPa,
the C state appears to persist down to the lowest measured temperatures. With
the aid of micromagnetic simulations, we propose that the driving mechanism
behind this change is a reduction in the magnetic anisotropy of the system, and
suggest that this could lead to an increase in stability of the skyrmion
lattice.
