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The transverse momentum $j_{\rm T}$-dependent jet fragmentation functions have been investigated in proton+proton (p $+$ p) and proton+lead (p $+$ Pb) collisions at $\sqrt{s_{\rm NN}} = 5.02~{\rm TeV}$ with a multiphase transport model containing both a simple quark coalescence mechanism and a new hybrid hadronization mechanism with coalescence and fragmentation processes. Hadronized by the new hadronization mechanism, the AMPT model achieves a quantitative description of the $j_{\rm T}$-dependent jet fragmentation functions measured by ALICE. Besides, no obvious jet-medium interaction and cold nuclear matter effects on the $j_{\rm T}$-dependent jet fragmentation functions in p $+$ Pb collisions were observed. We found the $j_{\rm T}$-dependent jet fragmentation functions are dominated by the quark coalescence contribution for the new hadronization mechanism, which can be decomposed into narrow and wide parts. The root mean square value of the wide part depends on the jet radius $R$ and jet transverse momentum $p_{\rm{T, jet}}$, which is sensitive to different hadronization mechanisms and their components. Therefore, the $j_{\rm T}$-dependent jet fragmentation functions are proposed as a sensitive probe to study the non-perturbative hadronization effect of jets in small colliding systems.