The rice–wheat rotation is the dominant cropping system in Bihar, where food security of the rural population depends heavily on the production of rice and wheat. In Bihar, farmers plant rice after the first significant rains, and climatic shocks induced by low temperatures and terminal heat stress at the end of the corresponding season can significantly affect rice and wheat yields. The present work evaluates the benefit of using an earlier date for planting rice, following the monsoon onset, in reducing thermal stress on rice–wheat systems. High-resolution gridded crop simulations using the APSIM model were performed to simulate potential yields using the monsoon onset and the farmers’ practice as planting dates. The monsoon onset was calculated using an agronomic definition, and farmers’ practice dates were estimated using satellite data. The results were analyzed in terms of planting dates, yields, and the incidence of temperature stress on rice and wheat by means of the APSIM yields limiting factors. The results show that the rice planting and harvest dates using the monsoon onset are, in general, 20–30 days earlier, which translates into higher and more stable potential yields, which can be up to 50% higher in wheat and 29% in rice. The incidence of thermal stress can be, on average, 12% lower in rice and 25% in wheat. These results can help design mitigation strategies for the impacts of temperature-induced shock events in the context of the advances in sub-seasonal and seasonal forecasting, targeting climate services for farmers in Bihar.