We investigate the semi-inclusive productions of spin-3/2 hadrons in unpolarized $e^+ e^-$ annihilation and deep inelastic lepton-nucleon scattering. The quark transverse momentum dependent (TMD) fragmentation functions (FFs) to spin-3/2 hadrons are defined for the first time from the decomposition of the quark-quark correlator at leading twist, 14 of which are newly defined for rank-3 tensor polarized hadron states. We perform a leading order calculation of the differential cross sections. For two-hadron production in $e^+ e^-$ annihilation, half of the 48 structure functions are found nonzero even if the spin of the second hadron is not analyzed, and ten of the rank-3 tensor polarized TMD FFs contribute. We also apply these newly defined FFs in semi-inclusive deep inelastic scattering (SIDIS) for the study of nucleon structures. For a polarized lepton beam, one third of 96 structure functions have nonzero leading order contributions and 42 of rank-3 tensor polarized TMD FFs contribute. For the polarized lepton, half of 192 structure functions are nonzero and 14 of them are from rank-3 tensor polarized hadron states.