Microseismic data and post-fracturing production analysis have confirmed a positive correlation between fracture complexity and production enhancement in fractured wells. While operators are looking for modifications in their pumping schedules such as adding pumping hesitations to enhance fracture complexity, the physics behind the effectiveness of these trial and error efforts is not fully understood. In this paper, we try to investigate different scenarios that may enhance fracture complexity by introducing changes in the pumping schedule. Since field evidence shows effectiveness of these techniques more in the presence of natural fractures, we first present a comprehensive workflow to model hydraulic fracturing by accounting for interactions with numerous natural fractures. This model is a coupled fluid flow and deformation finite element model with adaptive insertion of three-dimensional cohesive elements to simulate fracture propagation through the rock matrix and a network of intersecting natural fractures. Simulations have been implemented for different natural fracture patterns. Our analyses have shown that in addition to the differential stress and fractures' intersection angle, pumping rate and its breaks can play a significant role in fracture branching and its diversion into natural fractures. For continuously crosscutting natural fractures, higher injection rates are found to have a positive effect in overcoming the resistance of natural fractures in different directions. While in the case of discontinuously hierarchical natural fractures, this improvement can be very limited, while adding a hesitation in the middle of the pumping period can force the fractures to divert into other directions, which is effective to develop more complex fractures for different natural fracture patterns.