  New experimental data are presented on the scan rate dependence of the
magnetization hysteresis width $\Delta M(H)$ ($\propto$ critical current
density $J_c(H)$) in isothermal $M-H$ scans in a weakly pinned single crystal
of Ca$_3$Rh$_4$Sn$_{13}$, which displays second magnetization peak (SMP)
anomaly as distinct from the peak effect (PE). We observe an interesting
modulation in the field dependence of a parameter which purports to measure the
dynamical annealing of the disordered bundles of vortices injected through the
sample edges towards the destined equilibrium vortex state at a given $H$.
These data, in conjunction with the earlier observations made while studying
the thermomagnetic history dependence in $J_c(H)$ in the tracing of the minor
hysteresis loops, imply that the partially disordered state heals towards the
more ordered state between the peak field of the SMP anomaly and the onset
field of the PE. The vortex phase diagram in the given crystal of
Ca$_3$Rh$_4$Sn$_{13}$ has been updated in the context of the notion of the
phase coexistence of the ordered and disordered regions between the onset field
of the SMP anomaly and the spinodal line located just prior to the
irreversibility line. A multi-critical point and a critical point in the
($H,T$) region of the Bragg glass phase have been marked in this phase diagram
and the observed behaviour is discussed in the light of recent data on
multi-critical point in the vortex phase diagram in a single crystal of Nb.
