  Motivated by the recent observation of coherent elastic neutrino-nucleus
scattering (CE$\nu $NS) at the COHERENT experiment, our goal is to explore its
potential in probing important nuclear structure parameters. We show that the
recent COHERENT data offers unique opportunities to investigate the neutron
nuclear form factor. Our present calculations are based on the deformed Shell
Model (DSM) method which leads to a better fit of the recent CE$\nu $NS data,
as compared to known phenomenological form factors such as the Helm-type,
symmetrized Fermi and Klein-Nystrand. The attainable sensitivities and the
prospects of improvement during the next phase of the COHERENT experiment are
also considered and analyzed in the framework of two upgrade scenarios.
