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→aa→(bb¯)(bb¯), using proton-proton collision data collected by the ATLAS detector at the Large Hadron Collider at center-of-mass energy √s=13 TeV. This search focuses on the range 15 GeV≤m_a≤30 GeV, where the decay products are collimated; it is complementary to a previous search in the same final state targeting the range 20 GeV≤m_a≤60 GeV, where the decay products are well separated. A novel strategy for the identification of the a→bb¯ 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→aa→(bb¯)(bb¯), where the Higgs boson is produced in association with a Z boson.