The hydrodynamic theory is a powerful tool to study the nonlocal effects in metallic nanostructures that are too small to obey classical electrodynamics while still too large to be handled with a full quantum-mechanical theory. The existing hydrodynamic model can give accurate quantitative predictions for the plasmonic resonance shifts in metallic nanoplasmonics, yet is not able to predict the spectral width which is usually taken as a pre-set value instead. By taking account the fact that due to electron density spill-out from a surface, the Coulomb interaction screening is less efficient close the surface thus leads to a higher electron-electron scattering rate in this paper, we study how the electron-density-related damping rate induced by such Coulomb interaction will affect the plasmonic spectral broadening. We perform the simulation on a Na nanowire, which shows that the absorption spectra width is wider when the size of the nanowire becomes smaller. This result is consistent well with the reported experiment. Therefore, our theoretical model extends the existing hydrodynamic model and can provide much more quantum insight about nonlocal effects in metallic nanostructures.