  CeRhIn$_5$ is a prototypical antiferromagnetic heavy-fermion compound, whose
behavior in a magnetic field is unique. A magnetic field applied in the basal
plane of the tetragonal crystal structure induces two additional phase
transitions. When the magnetic field is applied along, or close to, the $c$
axis, a new phase characterized by a pronounced in-plane electronic anisotropy
emerges at $B^* \approx$ 30 T, well below the critical field, $B_c \simeq$ 50
T, to suppress the antiferromagnetic order. The exact origin of this new phase,
originally suggested to be an electronic-nematic state, remains elusive. Here
we report low-temperature specific-heat measurements in CeRhIn$_5$ in high
static magnetic fields up to 36 T applied along both the $a$ and $c$ axes. For
fields applied along the $a$ axis, we confirmed the previously suggested phase
diagram, and extended it to higher fields. This allowed us to observe a triple
point at $\sim$ 30 T, where the first-order transition from an incommensurate
to commensurate magnetic structure merges into the onset of the second-order
antiferromagnetic transition. For fields applied along the $c$ axis, we
observed a small but distinct anomaly at $B^*$, which we discuss in terms of a
possible field-induced transition, probably weakly first-order. We further
suggest that the transition corresponds to a change of magnetic structure. We
revise magnetic phase diagrams of CeRhIn$_5$ for both principal orientations of
the magnetic field based entirely on thermodynamic anomalies.
