  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, $H \rightarrow aa \rightarrow (b\bar{b})(b\bar{b})$, using
proton-proton collision data collected by the ATLAS detector at the Large
Hadron Collider at center-of-mass energy $\sqrt{s}=13$ TeV. This search focuses
on the regime where the decay products are collimated and in the range $15 \leq
m_a \leq 30$ GeV and is complementary to a previous search in the same final
state targeting the regime where the decay products are well separated and in
the range $20 \leq m_a \leq 60$ GeV. A novel strategy for the identification of
the $a \rightarrow b\bar{b}$ decays is deployed to enhance the efficiency for
topologies with small separation angles. The search is performed with 36
fb$^{-1}$ of integrated luminosity collected in 2015 and 2016 and sets upper
limits on the production cross-section of $H \rightarrow aa \rightarrow
(b\bar{b})(b\bar{b})$, where the Higgs boson is produced in association with a
$Z$ boson.
