Enabled by hybrid grant-based (GB) and grant-free (GF) transmission techniques, GF users of internet of things (IoT) devices and massive machine-type communications (mMTC) meet opportunities to share wireless resources with GB users. In this paper, we propose a rate splitting multiple access (RSMA) strategy for an emerging semi-grant-free (SGF) transmission system to increase connectivity and reliability. In the proposed RSMA assisted SGF (RSMA-SGF) scheme, the GF users apply the rate splitting principle to realize distributed contentions and utilize transmit power most effectively for robust transmissions, meanwhile keeping themselves transparent to the GB user. Compared to existing non-orthogonal multiple access (NOMA) assisted SGF schemes, the RSMA-SGF scheme significantly decreases outage probability and achieves full multiuser diversity gain without restricting the GB and GF users' target rates to a limited value region. Exact expressions and asymptotic analysis for the outage probability are provided to facilitate the system performance evaluation of the proposed RSMA-SGF scheme. Computer simulation results clarify the superior outage performance of the RSMA-SGF scheme and verify the accuracy of the developed analytical results.