  Superatomic crystal is a class of hierarchical materials composed of
atomically precise clusters assembled via van der Waals or covalent-like
interactions. AuTe$_2$Se$_{4/3}$, an all-inorganic superatomic superconductor
exhibiting superatomic-charge-density-wave (S-CDW), provides a first platform
to study the response of their collectively quantum phenomenon to the external
pressure in superatomic crystals. We reveal a competition between S-CDW and
superconductivity using cutting-edge measurements on thin flakes at low
pressures. Prominently, the pressure modulation of S-CDW ordering is 1$\sim$2
order of magnitudes (0.1 GPa) lower than that of conventional atomic
superconductors. As pressure increases to 2.5 GPa, the $T_{\mathrm{CDW}}$ is
suppressed and the superconducting transition temperature ($T_{\mathrm{c}}$) is
firstly enhanced, and reaches the maximum then quenches with increasing
pressure. Above 7.3 GPa, a second superconducting phase emerges, and then a
three-fold enhancement in the transition temperature ($T_{\mathrm{c}}$)
happens. Analyses of the crystal structure and theoretical calculations suggest
a pressure-mediated switch of the conduction channel from the $a$- to the
$b$-axis occur, followed by a dimensional crossover of conductivity and the
Fermi surface from 2D to 3D.
