A number of feedbacks regulate the response of Arctic sea ice to local atmospheric warming. Using a realistic coupled ocean‐sea ice model and its adjoint, we isolate a mechanism by which significant ice growth at the end of the melt season may occur as a lagged response to Arctic atmospheric warming. A series of perturbation simulations informed by adjoint model‐derived sensitivity patterns reveal the enhanced ice growth to be accompanied by a reduction of snow thickness on the ice pack. Detailed analysis of ocean‐ice‐snow heat budgets confirms the essential role of the reduced snow thickness for persistence and delayed overshoot of ice growth. The underlying mechanism is a snow‐melt‐conductivity feedback, wherein atmosphere‐driven snow melt leads to a larger conductive ocean heat loss through the overlying ice layer. Our results highlight the need for accurate observations of snow thickness to constrain climate models and to initialize sea ice forecasts. Plain Language Summary In this study we explore the relationship between Arctic sea ice growth and near‐surface air temperature in a modeling framework. By mapping the time‐ and space‐evolving sensitivity of the Arctic ice volume to changes in air temperature, we show that warming at the end of the melting season can lead to thicker ice at the end of the following winter. Warmer air temperatures can melt snow and remove the insulation it provides, exposing the ice surface to subfreezing air temperatures. We show that removal of this insulating snow layer is essential for enhancing sea ice growth later on, by allowing more heat to be conducted up and out of the underlying ocean, supporting seawater freezing. Our results highlight the importance of measuring snow thickness for accurately forecasting Arctic sea ice. Key Points Arctic sea ice sensitivity to atmospheric warming is computed using a numerical adjoint, revealing warming can lead to lagged sea ice growth Snow melt—in response to early warming—is the key preconditioner, enabling large conductive ocean heat loss to support lagged ice growth