An insert assembly is outfitted upon a lidded beverage container for enabling the user to transfer heat from a relatively hot assembly-contained beverage so as to prevent scalding prior to consumption. The insert assembly comprises a damming structure and a rim-engaging structure. The damming structure is formed from a semi-rigid material and is sized and shaped for receipt within the upper rim perimeter of a container structure. The rim-engaging structure is formed from a flexible material and extends outwardly from the beverage-damming structure. The rim-engaging structure is received intermediate the upper container rim and a lid for defining beverage-containing and beverage-cooling compartments. The insert structure comprises first and second apertures for outletting beverage and inletting air intermediate the beverage-containing and beverage-cooling compartments. The beverage-cooling compartment receives heat from the beverage received therewithin and thereby enables the beverage to cool before being outlet via a primary outlet of a lid.