  The heavy-fermion superconductor CeCoIn$_5$ displays an additional transition
within its superconducting (SC) state, whose nature is characterized by
high-precision studies of the isothermal field dependence of the entropy,
derived from combined specific heat and magnetocaloric effect measurements at
temperatures $T\geq 100$ mK and fields $H\leq 12$ T aligned parallel,
perpendicular and $18^\circ$ off the tetragonal [100] direction. For any of
these conditions, we do not observe an additional entropy contribution upon
tuning at constant temperature by magnetic field from the homogeneous SC into
the presumed Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) SC state. By contrast, for
$H\parallel [100]$ a negative isothermal entropy contribution, compatible with
spin-density-wave (SDW) ordering, is found. Our data exclude the formation of a
FFLO state in CeCoIn$_5$ for out-of-plane field directions, where no SDW order
exists.
