Spin distributions for various evaporation residues populated via complete and incomplete fusion of $^{16}\mathrm{O}$ with $^{124}\mathrm{Sn}$ at $6.3$ MeV/nucleon have been measured, using charged particles ($Z=1,2$)-$\ensuremath{\gamma}$ coincidence technique. Experimentally measured spin distributions of the residues produced as incomplete fusion products associated with ``fast'' $\ensuremath{\alpha}$- and $2\ensuremath{\alpha}$-emission channels observed in the ``forward cone'' are found to be distinctly different from those of the residues produced as complete fusion products. Moreover, ``fast'' $\ensuremath{\alpha}$-particles that arise from larger angular momentum in the entrance channel are populated at relatively higher driving input angular momentum than those produced through complete fusion. The incomplete fusion residues are populated in a limited, higher-angular-momentum range, in contrast to the complete fusion products, which are populated over a broad spin range.