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→ℓ^{+}ℓ^{-} (ℓ=e, μ). 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 TeV at the Large Hadron Collider and correspond to an integrated luminosity of 36.1±0.8 fb^{-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 cτ 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 cτ 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.