We study nonsingular cosmological scenarios in a general D-dimensional string effective action of the dilaton-modulus-axion system in the presence of the matter source. In the standard dilatonic Brans-Dicke parameter (ω = −1) with radiation, we analytically obtain singularity-free bouncing solutions where the universe starts out in a state with a finite curvature and evolves toward the weakly coupled regime. We apply this analytic method to the string-gas cosmology including the massive state in addition to the leading massless state (radiation), with and without the axion. We numerically find bouncing solutions which asymptotically approach an almost radiation-dominant universe with a decreasing curvature irrespective of the presence of the axion, implying that inclusion of the matter source is crucial for the existence of such solutions for ω = −1. In the theories with ω 6= −1, it is possible to obtain complete regular bouncing solutions with a finite dilaton and curvature in both past and future asymptotics for the general dimension, D. We also discuss the case where dilatonic higher-order corrections are involved to the tree-level effective action and demonstrate that the presence of axion/modulus fields and the matter source does not significantly affect the dynamics of the dilaton-driven inflation and the subsequent graceful exit.