This Letter presents a search for the production of a long-lived neutral particle (${Z}_{d}$) decaying within the ATLAS hadronic calorimeter, in association with a standard model (SM) $Z$ boson produced via an intermediate scalar boson, where $Z\ensuremath{\rightarrow}{\ensuremath{\ell}}^{+}{\ensuremath{\ell}}^{\ensuremath{-}}$ ($\ensuremath{\ell}=e$, $\ensuremath{\mu}$). The data used were collected by the ATLAS detector during 2015 and 2016 $pp$ collisions with a center-of-mass energy of $\sqrt{s}=13\text{ }\text{ }\mathrm{TeV}$ at the Large Hadron Collider and correspond to an integrated luminosity of $36.1\ifmmode\pm\else\textpm\fi{}0.8\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. No significant excess of events is observed above the expected background. Limits on the production cross section of the scalar boson times its decay branching fraction into the long-lived neutral particle are derived as a function of the mass of the intermediate scalar boson, the mass of the long-lived neutral particle, and its $c\ensuremath{\tau}$ from a few centimeters to one hundred meters. In the case that the intermediate scalar boson is the SM Higgs boson, its decay branching fraction to a long-lived neutral particle with a $c\ensuremath{\tau}$ approximately between 0.1 and 7 m is excluded with a 95% confidence level up to 10% for ${m}_{{Z}_{d}}$ between 5 and 15 GeV.