Small nuclear collisions are mainly sensitive to cold-nuclear-matter effects; however, the collective behavior observed in these collisions shows a hint of hot-nuclear-matter effects. The identified-particle spectra, especially the $\phi$ mesons which contain strange and antistrange quarks and has a relatively small hadronic-interaction cross section, are a good tool to study these effects. The PHENIX experiment has measured $\phi$ mesons in a specific set of small collision systems $p$$+$Al, $p$$+$Au, and $^3$He$+$Au, as well as $d$$+$Au [Phys. Rev. C 83, 024909 (2011)], at $\sqrt{s_{_{NN}}}=200$ GeV. The transverse-momentum spectra and nuclear-modification factors are presented and compared to theoretical-model predictions. The comparisons with different calculations suggest that quark-gluon plasma may be formed in these small collision systems at $\sqrt{s_{_{NN}}}=200$ GeV. However, the volume and the lifetime of the produced medium may be insufficient for observing strangeness-enhancement and jet-quenching effects. The comparison with calculations suggests that the main production mechanisms of $\phi$ mesons at midrapidity may be different in $p$$+$Al versus $p/d/^3$He$+$Au collisions at $\sqrt{s_{_{NN}}}=200$ GeV. While thermal quark recombination seems to dominate in $p/d/^3$He$+$Au collisions, fragmentation seems to be the main production mechanism in $p$$+$Al collisions.