This paper describes a search for beyond the Standard Model decays of the Higgs boson into a pair of new spin-0 particles subsequently decaying into $b$-quark pairs, $H\ensuremath{\rightarrow}aa\ensuremath{\rightarrow}(b\overline{b})(b\overline{b})$, using proton-proton collision data collected by the ATLAS detector at the Large Hadron Collider at center-of-mass energy $\sqrt{s}=13\text{ }\text{ }\mathrm{TeV}$. This search focuses on the range $15\text{ }\text{ }\mathrm{GeV}\ensuremath{\le}{m}_{a}\ensuremath{\le}30\text{ }\text{ }\mathrm{GeV}$, where the decay products are collimated; it is complementary to a previous search in the same final state targeting the range $20\text{ }\text{ }\mathrm{GeV}\ensuremath{\le}{m}_{a}\ensuremath{\le}60\text{ }\text{ }\mathrm{GeV}$, where the decay products are well separated. A novel strategy for the identification of the $a\ensuremath{\rightarrow}b\overline{b}$ decays is deployed to enhance the efficiency for topologies with small separation angles. The search is performed with $36\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of integrated luminosity collected in 2015 and 2016 and sets upper limits on the production cross section of $H\ensuremath{\rightarrow}aa\ensuremath{\rightarrow}(b\overline{b})(b\overline{b})$, where the Higgs boson is produced in association with a $Z$ boson.