The solar wind kinetic energy fueling all dynamical processes within the near-Earth space is extracted in a dynamo process at the magnetopause. This direct energy transfer from the solar wind into the magnetosphere depends on the orientation of the interplanetary magnetic field (IMF) as well as other solar wind parameters, such as the IMF magnitude and solar wind velocity. Using the GUMICS-4 magnetohydrodynamic (MHD) simulation, we find that the energy input from the solar wind into the magnetosphere depends on this direct driving as well as the magnetopause magnetic properties and their time history in such a way that the energy transfer can continue even after the direct driving conditions turned unfavorable. Such a hysteresis effect introduces discrepancies between the energy input proxies and the energy input measured from GUMICS-4, especially after strong driving, although otherwise the simulation energy input captures the system dynamics. For the cause of the effect, we propose a simple feedback mechanism based on magnetic flux accumulation in the tail lobes. By ideal MHD theory, the energy conversion at the magnetopause is proportional to the product of normal and tangential magnetic fields, the magnetic stress. During large magnetic flux accumulation, the tangential field at the magnetopause strengthens, enhancing the local instantaneous energy conversion and transfer. Our simulations show that this mechanism supports the energy transfer even under weak driving followed by favorable solar wind conditions and transfer up to 50% more power than without the feedback.