Properties of the underlying-event in pp interactions are investigated primarily via the strange hadrons $K_{S}^{0}$, $\Lambda $ and $\bar{\Lambda }$, as reconstructed using the ATLAS detector at the LHC in minimum-bias pp collision data at $\sqrt{s} = 13$ TeV. The hadrons are reconstructed via the identification of the displaced two-particle vertices corresponding to the decay modes , $\Lambda \rightarrow \pi ^-p$ and $\bar{\Lambda }\rightarrow \pi ^+\bar{p}$. 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 $10 < p_T \le 40$ GeV are studied using the number of prompt charged particles in the transverse region. The ratio $N(\Lambda + \bar{\Lambda })/N(K_{S}^{0})$ 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.