In order to ensure customer satisfaction, ride-hailing providers typically base their routing strategies around minimizing the total traffic delay faced by their users. However, accurately modeling the provider's routing problem is difficult, as realistic traffic networks are non-linear, stochastic, and time-varying. We approach this problem by modelling the dynamic traffic network using the Lighthill, Whitham, and Richards model and use stochastic path integral control to construct the routing strategy of the ride-hailing provider. Differing from previous results, we allow for multiple input and output locations, as well as varying speed limits and maximum traffic densities. Furthermore, we allow the ride-hailing provider to additionally control its traffic flow at splits in the traffic network, which avoids an exponential blow-up in the state space. A numerical example for a representative traffic network is provided to demonstrate the efficacy of the proposed method.