Stir 3 cups apple cider, ginger, cinnamon stick, and cloves together in a saucepan; bring to a boil, reduce heat to low, and simmer until slightly reduced, about 15 minutes.
Remove saucepan from heat and immediately stir brown sugar and honey into the cider mixture until dissolved into the liquid.
Pour the liquid into a wide glass dish and refrigerate until chilled completely, at least 45 minutes.
Mix the chilled apple cider mixture with the remaining apple cider and the pumpkin puree in a large pitcher; stir until the puree dissolves into the liquid.
