•The boundary conditions of Fu's equations in Budyko Framework are highlighted.•Equilibrium evaporation coefficient was adjusted by Fu's equations' interrelation.•We improved the complementary principle function for better evaporation estimate.•Our blended function substantially enhances annual E estimation performance.•The improved method shows the highest improvement of daily E in extreme climates. The complementary principle function presents a framework for estimating evaporation but faces long-term challenges in calibration with a free parameter-equilibrium evaporation coefficient. The blending the evaporation precipitation ratio with complementary principle function is proposed to provide a physical basis for estimation of the equilibrium evaporation coefficient within the Budyko framework; however, it overlooks the limited possible maximum evaporation and precipitation condition from Fu’s equations. In this study, we expanded the numerical ambiguity boundary conditions of Fu’s equations and introduced a new parameter γc into the blended function to fully capture the environmental conditions, aiming to mitigate errors related to extreme climates. The result indicated our improved blended function performed better than the original function. Compared to the original function, the estimated annual average evaporation across stations improved from 0.12 to 0.57 in coefficient of determination (R2) and from 322.83 mm to 172.47 mm in Root Mean Square Error. For daily evaporation estimation, the Nash-Sutcliffe Efficiency coefficient (NSE) increased from 0.32 to 0.47 and R2 increased from 0.38 to 0.48 in all validation sites. The modified blended method showed a better performance in most of the AmeriFLUX stations, especially with the highest improvement in extremely dry and wet sites. Our new approach represents a promising evaporation method in extreme climate conditions and provides support for future research to identify the rigorous equation constraints of Budyko equations.