An insert construction or formation is attachable to a basic lid construction for outfitting a hot beverage or liquid container assembly for enabling a user to enhance heat transfer from a relatively hot assembly-contained beverage or liquid prior to consumption. The insert construction or formation provides a damming structure located in adjacency to the primary liquid outlet of the basic lid construction for selectively transferring liquid intermediate a liquid-containing compartment and at least one beverage-cooling channel made part of the damming structure. Each beverage-cooling channel or formation directs liquid therethrough and transfers heat therefrom before the liquid exits the primary liquid outlet. The primary dam structure or insert construction thereby enables the user to redirect liquid movement via the basic lid construction for delaying liquid delivery via the primary beverage outlet and transferring heat therefrom prior to liquid consumption. Certain packaging methodology of the complex lid construction is further contemplated.