Quantum resistance is vital for emerging cryptographic systems as quantum technologies continue to advance towards large-scale, fault-tolerant quantum computers. Resistance may be offered by quantum key distribution (QKD), which provides information-theoretic security using quantum states of photons, but may be limited by transmission loss at long distances. An alternative approach uses classical means and is conjectured to be resistant to quantum attacks, so-called post-quantum cryptography (PQC), but it is yet to be rigorously proven, and its current implementations are computationally expensive. To overcome the security and performance challenges present in each, here we develop hybrid protocols by which QKD and PQC inter-operate within a joint quantum-classical network. In particular, we consider different hybrid designs that may offer enhanced speed and/or security over the individual performance of either approach. Furthermore, we present a method for analyzing the security of hybrid protocols in key distribution networks. Our hybrid approach paves the way for joint quantum-classical communication networks, which leverage the advantages of both QKD and PQC and can be tailored to the requirements of various practical networks.