In this study, we relax one of the general assumptions in the hub location literature by allowing routed flows between nonhub nodes. In hub networks, different flows are consolidated and routed via collection, interhub, and distribution arcs. Due to consolidation, some flows travel long paths despite closeness of their origin and destination. In this study, we allow direct flows by penalizing by a scalar factor of original cost of transshipment between these arcs. We present mathematical models for median, center, and set covering versions of the problem for single- and multi-allocation cases. We test the models with the CAB and TR data sets. We discuss the properties of established direct connections for different models by using another mathematical model where the number of direct flows is bounded and interpret the effect of changes in problem parameters.