  Influence of metamagnetism on superconductivity is studied for the field
along an easy $a$-axis by AC magnetic susceptibility, magnetization, and
magnetocaloric effect (MCE) in UTe$_2$. The improvement of crystal quality
leads to an upward curvature of $T_c$($B$) above 6 T, that results in a largely
enhanced upper critical field of 12 T. The entropy analysis also shows a
metamagnetic crossover around 6 T. The sharp negative peak in MCE below $T_c$
indicates that the crossover becomes a transition of first-order character in
the superconducting state. Our results shows that metamagnetism induces a
transition inside the superconducting phase and stabilizes the
superconductivity for the field along the easy axis.
