  We report Shubnikov-de Haas (SdH) oscillation measurements on FeSe under high
pressure up to $P$ = 16.1 kbar. We find a sudden change in SdH oscillations at
the onset of the pressure-induced antiferromagnetism at $P$ $\sim$ 8 kbar. We
argue that this change can be attributed to a reconstruction of the Fermi
surface by the antiferromagnetic order. The negative d$T_c$/d$P$ observed in a
range between $P$ $\sim$ 8 and 12 kbar may be explained by the reduction in the
density of states due to the reconstruction. The ratio of the transition
temperature to the effective Fermi energy remains high under high pressure:
$k_BT_c/E_F$ $\sim$ 0.1 even at $P$ = 16.1 kbar.
