Collisions of small systems show signatures suggestive of collective flow associated with QGP formation in heavy-ion collisions. Jet quenching is also a consequence of QGP formation, but no significant evidence of it in small systems has been found to date. Measuring or constraining the magnitude of jet quenching in small systems is essential to determine the limits of QGP formation. The ALICE Collaboration presents a broad search for jet quenching in minimum bias (MB) and high multiplicity (HM) pp and p-Pb collisions, based on several observables: the semi-inclusive acoplanarity distribution of jets recoiling from a high-$p_{\text{T}}$ hadron (h+jet) and intra-jet measurements. Marked broadening of the h+jet acoplanarity distribution is observed in HM compared to MB events, which could arise from jet quenching. Both data and PYTHIA simulations suggest that this broadening arises from a generic bias of the HM selection towards multi-jet final states. We also report the average charged-particle multiplicity and jet fragmentation functions in HM and MB-selected populations in pp and p-Pb collisions, whose differences are qualitatively described by PYTHIA. Finally, to disentangle jet fragmentation and hadronisation effects, the transverse momentum ($j_{\rm T}$) distributions of charged-particle jet constituents are measured for several longitudinal momentum fraction of jet constituents ($z$) ranges in pp and p-Pb collisions. No significant difference is observed within the measurement uncertainties. We discuss the implications of these results for the understanding of collective effects and jet quenching in small systems.