  We provide thermodynamic evidence for the presence of a
magnetic-field-induced small phase pocket near the antiferro-quadrupole (AFQ)
phase boundary in the non-Kramers $\Gamma_3$ doublet system PrIr$_2$Zn$_{20}$.
In particular, we measured the specific heat as functions of temperature $T$,
magnetic field $B$, and field angle $\phi_B$, and found a second specific-heat
anomaly in a relatively wide field-angle range near $B \parallel [001]$,
although fine tuning of the field strength is required. We also investigated
the rotational magnetocaloric effect and evaluated an entropy change in this
phase pocket. The present findings demonstrate that multipole degrees of
freedom give rise to a magnetic-field-induced exotic order in
PrIr$_2$Zn$_{20}$, suggesting the possibility of switching between the order
parameters or emergence of a multiple-$q$ order of quadrupoles.
