Properties of the underlying-event in pp interactions are investigated primarily via the strange hadrons K0S, Λ and ¯Λ, as reconstructed using the ATLAS detector at the LHC in minimum-bias pp collision data at √s=13 TeV. The hadrons are reconstructed via the identification of the displaced two-particle vertices corresponding to the decay modes , Λ→π−p and ¯Λ→π+¯p. These are used in the construction of underlying-event observables in azimuthal regions computed relative to the leading charged-particle jet in the event. None of the hadronisation and underlying-event physics models considered can describe the data over the full kinematic range considered. Events with a leading charged-particle jet in the range of 10<pT≤40 GeV are studied using the number of prompt charged particles in the transverse region. The ratio N(Λ+¯Λ)/N(K0S) as a function of the number of such charged particles varies only slightly over this range. This disagrees with the expectations of some of the considered Monte Carlo models.