  In this study, we attempt to introduce a model to illustrate neutrino
phenomenology by incorporating two right-handed fermion triplet superfields,
i.e., $\Sigma_{R_j}$, in the presence of the modular symmetry $\Gamma_3^\prime
\simeq A_4^\prime$, a double cover of the $A_4$ modular symmetry. The
motivation in utilizing double cover is, so far only even modular forms were
considered for constructing modular invariant models, but, in this case, it is
possible to extend the modular invariance approach to general integral weight
modular forms, i.e., the odd weight modular forms. Hence, this type of
amalgamation between $T^\prime$ modular symmetry and minimally extending the
seesaw can correctly explain the neutrino phenomenology. Additionally, we have
made an attempt to accommodate the most recent measurement of the $W$ boson
mass, published by the CDF-II collaboration and shed some light on the recent
results of muon $(g-2)$. Finally, we have discussed lepton flavor violation in
order to establish a constraint on the mass of right-handed fermion.
