  We present the magnetic and structural properties of
[Cu(pyrazine)$_{0.5}$(glycine)]ClO$_4$ under applied pressure. As previously
reported, at ambient pressure this material consists of quasi-two-dimensional
layers of weakly coupled antiferromagnetic dimers which undergo Bose-Einstein
condensation of triplet excitations between two magnetic field-induced quantum
critical points (QCPs). The molecular building blocks from which the compound
is constructed give rise to exchange strengths that are considerably lower than
those found in other $S = 1/2$ dimer materials, which allows us to determine
the pressure evolution of the entire field-temperature magnetic phase diagram
using radio-frequency magnetometry. We find that a distinct phase emerges above
the upper field-induced transition at elevated pressures and also show that an
additional QCP is induced at zero-field at a critical pressure of $p_{\rm c} =
15.7(5)$ kbar. Pressure-dependent single-crystal X-ray diffraction and density
functional theory calculations indicate that this QCP arises primarily from a
dimensional crossover driven by an increase in the interdimer interactions
between the planes. While the effect of quantum fluctuations on the lower
field-induced transition is enhanced with applied pressure, quantum Monte Carlo
calculations suggest that this alone cannot explain an unconventional asymmetry
that develops in the phase diagram.
