  A fundamental problem posed from the study of correlated electron compounds,
of which heavy-fermion systems are prototypes, is the need to understand the
physics of states near a quantum critical point (QCP). At a QCP, magnetic order
is suppressed continuously to zero temperature and unconventional
superconductivity often appears. Here, we report pressure ($P$) -dependent
$^{115}$In nuclear quadrupole resonance (NQR) measurements on heavy-fermion
antiferromagnet CeRh$_{0.5}$Ir$_{0.5}$In$_5$. These experiments reveal an
antiferromagnetic (AF) QCP at $P_{\rm c}^{\rm AF}$ = 1.2 GPa where a dome of
superconductivity reaches a maximum transition temperature $T_{\rm c}$.
Preceding $P_{\rm c}^{\rm AF}$, however, the NQR frequency $\nu_{\rm Q}$
undergoes an abrupt increase at $P_{\rm c}^{\rm *}$ = 0.8 GPa in the
zero-temperature limit, indicating a change from localized to itinerant
character of cerium's $f$-electron and associated small-to-large change in the
Fermi surface. At $P_{\rm c}^{\rm AF}$ where $T_{\rm c}$ is optimized, there is
an unusually large fraction of gapless excitations well below $T_{\rm c}$ that
implicates spin-singlet, odd-frequency pairing symmetry.
