This Letter presents the results of an investigation into the controlling influence of large-scale magnetic field of the Sun in determining the solar wind outflow using two magnetostatic coronal models: current sheet source surface (CSSS) and potential field source surface. For this, we made use of the Wang and Sheeley inverse correlation between magnetic flux expansion rate (FTE) and observed solar wind speed (SWS) at 1 au. During the period of study, extended over solar cycle23 and beginning of solar cycle24, we found that the coefficients of the fitted quadratic equation representing the FTE-SWS inverse relation exhibited significant temporal variation, implying the changing pattern of the influence of FTE on SWS over time. A particularly noteworthy feature is an anomaly in the behavior of the fitted coefficients during the extended minimum, 2008-2010 (CRs 2073 -2092, which is considered due to the particularly complex nature of the solar magnetic field during this period. However, this variation was significant only for the CSSS model, though not a systematic dependence on the phase of the solar cycle. Further, we noticed that the CSSS model demonstrated better solar wind prediction during the period of study, which we attribute to the treatment of volume and sheet currents throughout the corona and the more accurate tracing of footpoint locations resulting from the geometry of the model.