  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.
