What is the appropriate action to take to prevent a casualty from getting chilled after immobilizing their leg?

Covering the casualty to keep them from getting chilled is the appropriate action following the immobilization of their leg. This measure is crucial because exposure to cold can exacerbate the casualty's condition, potentially leading to hypothermia. By covering the casualty, warmth is retained, and the risk of temperature loss is minimized.

This practice recognizes the importance of body temperature regulation in injured individuals. Even if the injury itself does not directly involve exposure to colder environments, the stress of the injury and possible shock can lead to vulnerability to cold conditions. Therefore, providing insulation through covering can significantly contribute to the casualty's overall care and comfort.

In contrast, other actions like loosening binding clothing, while potentially helpful in some situations, do not specifically address the immediate concern of preventing chilling. Administering pain medication does not directly relate to temperature control, and moving the casualty to a warmer place might not always be feasible in the immediate context, especially if it risks aggravating their injury or if immediate transport is not possible. Thus, the most effective immediate action is to cover the casualty to retain their body heat.