We study the magnetocaloric metamagnetic transition in LaFe11.74Mn0.06Si1.20 and LaFe11.76Mn0.06Si1.18H1.65 under hydrostatic pressure up to 1.2 GPa. For both compounds, hydrostatic pressure depresses the zero field critical temperature. However, in detail, pressure influences the magnetic properties in different ways in the two compounds. In the dehydrogenated case the transition broadens under pressure whereas in the hydrogenated case the transition sharpens. In both cases thermal hysteresis increases under pressure, although with different trends. These observations suggest both intrinsic and extrinsic hysteresis loss brought about by the use of hydrostatic pressure. We explore the multicaloric field‐pressure cycle, demonstrating that although the gain introduced by overcoming the magnetic hysteresis loss is closely countered by the loss introduced in the pressure cycle, there are significant advantages in that the temperature range of operation can be finely tuned and extended, and the magnetocaloric transition can operate in lower absolute applied fields (<0.5 T), potentially overcoming one of the most significant bottlenecks to the commercialization of this technology.