  The fusion and transfer induced fission reaction $^{9}$Be($^{238}$U,~f) with
6.2 MeV/u beam energy, using a unique setup consisting of AGATA, VAMOS++ and
EXOGAM detectors, was used to populate through the fission process and study
the neutron-rich $^{119,121}$In isotopes. This setup enabled the prompt-delayed
$\gamma$-ray spectroscopy of isotopes in the time range of $100~\rm{ns} -
200~\mu\rm{s}$. In the odd-$A$ $^{119,121}$In isotopes, indications of a short
half-life $19/2^{-}$ isomeric state, in addition to the previously known
$25/2^{+}$ isomeric state, were observed from the present data. Further, new
prompt transitions above the $25/2^{+}$ isomer in $^{121}$In were identified
along with reevaluation of its half-life. The experimental data were compared
with the theoretical results obtained in the framework of large-scale
shell-model calculations in a restricted model space. The $\langle \pi g_{9/2}
\nu h_{11/2};I \arrowvert \hat{\mathcal{H}}\arrowvert \pi g_{9/2} \nu
h_{11/2};I\rangle$ two-body matrix elements of residual interaction were
modified to explain the excitation energies and the $B(E2)$ transition
probabilities in the neutron-rich In isotopes. The (i) decreasing trend of
$E(29/2^{+}) - E(25/2^{+})$ in odd-In (with dominant configuration $\pi
g_{9/2}^{-1}\nu h_{11/2}^{-2}$ and maximum aligned spin of $29/2^{+}$) and (ii)
increasing trend of $E(27/2^{+}) - E(23/2^{+})$ in odd-Sb (with dominant
configuration $\pi g_{7/2}^{+1}\nu h_{11/2}^{-2}$ and maximum aligned spin of
$27/2^{+}$) with increasing neutron number could be understood as a consequence
of hole-hole and particle-hole interactions, respectively.
