A particle-$\ensuremath{\gamma}$-coincidence experiment is performed to investigate the role of high $\ensuremath{\ell}$ values in the production of direct-$\ensuremath{\alpha}$-emitting channels (associated with incomplete fusion) in $^{12}\mathrm{C}+^{169}\mathrm{Tm}$ system. Spin distributions of various $\mathit{xn}/\mathit{pxn}/\ensuremath{\alpha}\mathit{xn}/2\ensuremath{\alpha}\mathit{xn}$ channels are measured at ${E}_{\mathrm{lab}}=5.6A$ and $6.5A$ MeV. Entirely different de-excitation patterns are observed in direct-$\ensuremath{\alpha}$-emitting channels and fusion-evaporation channels. The fusion-evaporation channels are found to be strongly fed over a broad spin range. While narrow range feeding for only high-spin states was observed in the case of direct-$\ensuremath{\alpha}$-emitting channels, in the present work, incomplete fusion is shown to be a promising tool to populate high-spin states in final reaction products. To have better insight into the associated $\ensuremath{\ell}$ values in different reaction channels, the present data are compared with similar data obtained in $^{16}\mathrm{O}({E}_{\mathrm{lab}}\ensuremath{\approx}5.6A \mathrm{MeV}) + ^{169}\mathrm{Tm}$ system. The mean driving angular momenta involved in the production of direct-$\ensuremath{\alpha}$-emitting channels are found to be higher than those involved in the production of fusion-evaporation channels. Direct-$\ensuremath{\alpha}$ multiplicity in the forward cone increases with driving angular momenta, which indicates the origin of direct-$\ensuremath{\alpha}$-emitting channels at high $\ensuremath{\ell}$ values in noncentral interactions.