Since long the measurement of strange and multi-strange hadron production rates has demonstrated to be a key measurement in the investigation of the properties of nuclear matter created in high energy collisions. In particular, the observation of strangeness enhancement in collisions of large systems, such as Pb-Pb, is historically considered as a signature of the formation of a QGP phase in such collisions. Recently, a relative increase of strange hadron production rates with respect to those of non-strange particles was also observed in high multiplicity pp and p-Pb collisions by the ALICE experiment. In this work, we present results of strange and multi-strange hadron measurements as a function of the charged particle multiplicity in pp collisions at 7 and 13 TeV. The dependence with the collision energy and comparisons with theoretical predictions based on QCD inspired model are discussed. Moreover, performance studies with a large sample of high-multiplicity triggered events in pp collisions at 13 TeV are also presented. This last data sample extends the reach in multiplicity currently available in the minimum bias sample, allowing for a better comparison with results from p-Pb and peripheral Pb-Pb collisions.