We investigate the single-spin asymmetry with a $\sin\phi_{S_h}$ modulation for the transversely polarized proton and $\Lambda$ production in semi-inclusive inelastic scattering process, where $\phi_{S_h}$ is the azimuthal angle of the transverse spin of the final hadron. Theoretically, the spin asymmetry can be interpreted by the convolution of the twist-3 transverse momentum dependent distributions and twist-2 fragmentation functions. In this work, three different origins in terms of the $h H_1$ term, the $f^\perp D_{1T}^\perp$ term and the $g^\perp G_{1T}$ term are taken into account simultaneously for this asymmetry.

We calculate the twist-3 quark transverse momentum dependent distributions $h$, $f^\perp$ and $g^\perp$ by using the quark spectator diquark model, and we investigate the role of the fragmentation functions $H_1$, $D_{1T}^\perp$ and $G_{1T}$ in the $\sin\phi_{S_h}$ asymmetry as well. We also predict the numerical results of the asymmetries for the proton and the $\Lambda$ production at JLab with a 12 $\mathrm{GeV}$ beam and at COMPASS with a 160 $\mathrm{GeV}$ beam, separately. From the comparison of the different sources for the asymmetry, we find that, the distribution $h$ and the fragmentation function $H_1$ give the dominant contribution to the $\sin\phi_{S_{h}}$ asymmetry for proton production, while the distribution $f^\perp$ might be probed by the convolution with $D_{1T}^\perp$ in the $\Lambda$ production at JLab 12 $\mathrm{GeV}$.