Semantic communication is envisioned as a promising solution for tackling the data transmission challenges in future wireless communication systems. In this paper, we propose a novel semantic importance-enhanced multi-hop semantic communication (SIMSC) framework that integrates semantic communication with a map-and-forward (MF) relaying scheme. To enhance transmission reliability, the framework employs a semantic importance (SI) calculation module to aid in training joint source-channel coding (JSCC) with relays, thereby better protecting the delivery of crucial semantic information over wireless channels with long-distance signal propagation. Different from conventional amplify-and-forward (AF) and decode-and-forward (DF) relaying schemes, the proposed MF scheme leverages a deep neural network (DNN) to adaptively transform the received signal, optimizing it for multi-hop transmission. This enhances robustness and flexibility in varying relay conditions. Numerical results demonstrate that the proposed SIMSC framework performs notably better than conventional solutions that do not utilize semantic coding or relays. Moreover, the MF relaying scheme outperforms the AF and DF schemes, especially when relay distances vary across hops.