The main goal of the ALICE experiment is to study the physics of strongly interacting matter, including the properties of the quark$-$gluon plasma (QGP). The increase in relative production of strange hadrons with respect to non-strange hadrons as a function of multiplicity is historically considered as one of the signatures of QGP formation during the evolution of the system created in heavy-ion collisions. Recent measurements performed in high-multiplicity proton$-$proton (pp) and proton$-$lead (p$-$Pb) collisions have shown features that are reminiscent of those observed in lead$-$lead (Pb$-$Pb) collisions. The microscopic origin of this phenomenon is still not fully understood: whether it is related to soft particle production or to hard scattering events, such as jets? To separate strange hadrons produced in jets from those produced in soft processes, the angular correlation between high-$p_{\mathrm{T}}\:$charged particles and strange hadrons has been exploited. The near-side jet yield and the out-of-jet yield of $\mathrm{K_S^0},\:\Xi,\:\Lambda$ and $\phi$ have been studied as a function of the multiplicity of charged particles produced in pp collisions at $\sqrt{s}$ = 5.02 TeV and 13 TeV and p$-$Pb collisions at $\sqrt{s_{\mathrm{NN}}}$ = 5.02 TeV. New results suggest that out-of-jet processes are the dominant contributor to strange particle production.