Heatwaves have devastating impacts on societies and ecosystems. Their frequencies and intensities are increasing globally with anthropogenic climate change. Statistical mod els using Extreme Value Theory (EVT) have been used for quantifying risks of extreme temperatures but recent very intense events have cast doubt on their ability to represent the tail probabilities of temperatures. Using outputs from a large ensemble of a climate model, we show that physics-based estimates of the upper-bound of temperatures in the mid-latitudes are 3–8°C higher than suggested by EVT-based models. We propose a new method to bridge the gap between the physical and statistical estimates by forcing the EVT-based models to have an upper bound coherent with the bound provided by the instability of the air column. We show that our method reduces the underestimation of tail risks while not deteriorating the performance of the statistical models on the core of the distribution of extreme temperatures.