Beat cream cheese, 1 tablespoon milk, and sugar together in a large bowl with a whisk until blended. Stir in half of whipped topping; spread into crust.
Whisk pumpkin puree, 1 cup milk, pudding mix, cinnamon, ginger, and ground cloves together in a bowl until mixture is thick and blended. Spread over cream cheese mixture.
Chill in the refrigerator until set, about 4 hours. Remove from the refrigerator; top with remaining whipped topping before serving.