Preheat the oven to 350º F. and lightly grease an 8 x 8-inch baking dish.
Cut the bread into 1-inch cubes and set aside.
Whisk together the pumpkin puree, cornstarch, sugar, spices, and salt in a large bowl.
Once smooth and thoroughly combined, add in the nondairy milk and optional rum, and stir until the mixture is homogeneous.
Toss in the bread cubes and crystallized ginger, and use a wide spatula to gently stir everything together.
Pour the mixture into the prepared pan and smooth it evenly across the bottom. Bake for 45 to 50 minutes. The edges should appear set, but the interior will remain quite soft and moist. Good bread pudding should be very moist, so be careful not to over bake it!
Let cool for about 15 minutes before serving. You can either take the time to make nice, neat slices like, or just grab a large spoon and scoop it out onto plates. Serve warm with ice cream, Caramel Sauce, or both.