  We have performed de Haas-van Alphen (dHvA) measurements of the heavy-fermion
superconductor CeCoIn$_5$ down to 2 mK above the upper critical field. We find
that the dHvA amplitudes show an anomalous suppression, concomitantly with a
shift of the dHvA frequency, below the transition temperature $T_{\rm n}=20$
mK. We suggest that the change is owing to magnetic breakdown caused by a
field-induced antiferromagnetic (AFM) state emerging below $T_{\rm n}$,
revealing the origin of the field-induced quantum critical point (QCP) in
CeCoIn$_5$. The field dependence of $T_{\rm n}$ is found to be very weak for
7--10 T, implying that an enhancement of AFM order by suppressing the critical
spin fluctuations near the AFM QCP competes with the field suppression effect
on the AFM phase. We suggest that the appearance of a field-induced AFM phase
is a generic feature of unconventional superconductors, which emerge near an
AFM QCP, including CeCoIn$_5$, CeRhIn$_5$, and high-$T_{\rm c}$ cuprates.
