We consider a call center that can offload (or co-source) some or all of its arriving customers to an external provider. These customers are impatient—in the sense that they will eventually abandon the queue if left unserved—and arrive according to a non-stationary but known arrival pattern during a finite time horizon. By co-sourcing customers, the call center can reduce delays and curtail customer abandonment, however for each customer the call center co-sources, the call center must pay the external provider a price based on the time of transfer. We view the resulting queueing system as a fluid model where the time horizon naturally separates into alternating overloaded, critically loaded, and underloaded time periods. We characterize the optimal proportion of co-sourced demand and the equilibrium co-sourcing prices, both as a functions of time. We find that there always exist “all or nothing” optimal co-sourcing proportions: We find that there always exist “all or nothing” optimal co-sourcing proportions: at any given point in time, unless the system is critically loaded, either all arriving customers should be co-sourced, or none of them should be co-sourced (i.e., they should all be served in-house). Moreover, we find that both extremes can exist in both overloaded and underloaded time intervals.