The equilibrium flow assignment problem is an optimization problem, which allows transportation engineers to estimate the usability and congestion on routes of an available network. Equilibration techniques appeared to be among the very first ideas on solving the problem due to their simplicity. Indeed, transferring pieces of flow between the available routes in order to shift a flow from a route with a longer period of travel time to a route with a shorter period of travel time seems to be quite a natural way for solving the equilibrium flow assignment problem. The approach demonstrates itself to be fruitful and easy to apply from a computational perspective. As a computational result, this method returns assignment patterns that provide equal travel times on used routes between origin-destination pairs. However, when we require the highest level of accuracy, we face multiple re-assignments of very small pieces of flow that leads to needs in additional computation efforts. In other words, the higher precision, the slower flow equilibration. In the present paper, we provide the computational results of a computational strategy that combines two equilibration procedures we developed for different data types. The first procedure obtains a fast but not so accurate result, while the second one uses the output of the first as the initial solution to achieve higher precision. We show that the combination of procedures demonstrates better performance compared to their separate usage. We believe our findings can give fresh insights to transportation engineers.