We study the single-spin asymmetries with the sin φS and sin(2φ h − φS) angular dependences for charged and neutral pions produced in semi-inclusive deep inelastic scattering on the transversely polarized proton target. The theoretical interpretations of the two asymmetries are presented in terms of the convolution of the twist-3 quark transverse momentum dependent distributions and twist-2 fragmentation functions. Specifically, we investigate the role of the distributions fT , hT and h ⊥ T in the sin φS asymmetry, as well as the role of the distributions f ⊥ T , hT and h ⊥ T in the sin(2φ h −φS) asymmetry. We calculate these distributions in a spectator-diquark model and predict the corresponding asymmetries for the first time, considering the kinematics at HERMES, JLab and COMPASS. The numerical estimates show that the asymmetries are sizable, and the dominant contribution to the sin φS asymmetry comes from the T-odd distribution fT , while f ⊥ T gives the main contribution to the sin(2φ h − φS) asymmetry. The future measurements on these asymmetries can shed light on the information of twist-3 transverse momentum dependent distributions.