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The effect of hydrostatic pressure on thermally and field-induced first-order magnetic phase transitions is studied in the La (Fe, Si) 13-type compounds. A peculiar series of consecutive field-induced transitions is realized using a distinct combination of hydrostatic pressure and negative pressure created by the interstitial insertion of hydrogen. The pressure-induced discontinuous magnetization jumps result in an enhanced cooling power, thus opening up the possibility to exploit in full the magnetocaloric potential of this compound class.