We studied the anisotropy of the superconducting upper critical field $H_{rm c2}$ in the heavy-fermion superconductor UTe$_2$ under hydrostatic pressure by magnetoresistivity measurements. In agreement with previous experiments we confirm that superconductivity disappears near a critical pressure $p_{rm c} approx 1.5$~GPa, and a magnetically ordered state appears. The unusual $H_{rm c2}(T)$ at low temperatures for $H parallel a$ suggests that the multiple superconducting phases which appear under pressure have quite different $H_{rm c2}$. For a field applied along the hard magnetization $b$ axis $H_{rm c2} (0)$ is glued to the metamagnetic transition $H_{rm m}$ which is suppressed near $p_{rm c}$. The suppression of $H_{rm m}$ with pressure follows the decrease of temperature $T_{chi}^{rm max}$, at the maximum in the susceptibility along $b$. The strong reinforcement of $H_{rm c2}$ at ambient pressure for $H parallel b$ above 16~T is rapidly suppressed under pressure due to the increase of $T_{rm sc}$ and the decrease of $H_{rm m}$. The change in the hierarchy of the anisotropy of $H_{rm c2}(0)$ on approaching $p_{rm c}$ points out that the $c$ axis becomes the hard magnetization axis.
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