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We present the magnetic and structural properties of [Cu(pyrazine)0.5(glycine)]ClO4 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 pc = 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.