This paper proposes the concept of adaptive multi-rate (AMR), which jointly employs switching between two links and adaptive rate on each link, for hybrid free-space optical/radio-frequency (FSO/RF) systems. Moreover, we present the cross-layer design of AMR switching, which is based on both the physical and link layers with an automatic-repeat request (ARQ) scheme. We develop an analytical framework based on a Markov chain model for system performance analysis. System performance metrics, including frame-error rate, goodput and link switching probability, are analytically studied over fading channels. Numerical results quantitatively show how the proposal significantly outperforms conventional ones with physical layer-based design and/or fixed-rate switching operation.