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CeOs₄Sb₁₂, a member of the skutterudite family, has an unusual semimetallic low-temperature$\mathcal{L}$-phase that inhabits a wedge-like area of the fieldH—temperatureTphase diagram. We have conducted measurements of electrical transport and megahertz conductivity on CeOs₄Sb₁₂single crystals under pressures of up to 3 GPa and in high magnetic fields of up to 41 T to investigate the influence of pressure on the differentH–Tphase boundaries. While the high-temperature valence transition between the metallic$\mathcal{H}$-phase and the$\mathcal{L}$-phase is shifted to higherTby pressures of the order of 1 GPa, we observed only a marginal suppression of the$\mathcal{S}$-phase that is found below 1 K for pressures of up to 1.91 GPa. High-field quantum oscillations have been observed for pressures up to 3.0 GPa and the Fermi surface of the high-field side of the$\mathcal{H}$-phase is found to show a surprising decrease in size with increasing pressure, implying a change in electronic structure rather than a mere contraction of lattice parameters. We evaluate the field-dependence of the effective masses for different pressures and also reflect on the sample dependence of some of the properties of CeOs₄Sb₁₂which appears to be limited to the low-field region.