This paper proposes a novel hybrid interweave-underlay spectrum access for a cognitive amplify-and-forward relay network where the relay forwards the signals of both the primary and secondary networks. In particular, the secondary network (SN) opportunistically operates in interweave spectrum access mode when the primary network (PN) is sensed to be inactive and switches to underlay spectrum access mode if the SN detects that the PN is active. A continuous-time Markov chain approach is utilized to model the state transitions of the system. This enables us to obtain the probability of each state in the Markov chain. Based on these probabilities and taking into account the impact of imperfect spectrum sensing of the SN, the probability of each operation mode of the hybrid scheme is obtained. To assess the performance of the PN and SN, we derive analytical expressions for the outage probability, outage capacity, and symbol error rate over Nakagami-m fading channels. Furthermore, we present comparisons between the performance of underlay cognitive cooperative radio networks (CCRNs) and the performance of the considered hybrid interweave-underlay CCRN in order to reveal the advantages of the proposed hybrid spectrum access scheme. Eventually, with the assistance of the secondary relay, performance improvements for the PN are illustrated by means of selected numerical results.