Atole will keep well refrigerated for two to three days, though it will thicken. To reheat, combine with a small amount of milk or water and stir over low heat until warmed through.
In a large heavy pot, combine the cubed pumpkin, piloncillo, spices and salt with 1/4 cup of water. Set over medium heat and stir frequently until the the piloncillo melts. Continue to cook, covered, until the pumpkin falls apart, 15 to 20 minutes. Puree in a blender or food processor until smooth.
In a large heavy pot, combine the water, milk and cinnamon stick and bring to a high simmer. Lower heat and whisk in the masa harina in a slow, steady stream, stirring constantly to eliminate any lumps. Add the pureed pumpkin and simmer until slightly thickened but still pourable, 5 to 10 minutes. Serve warm with added sugar to taste, and garnish with candied ginger.