Heat the oil in a large skillet over medium heat and brown the Andouille about 3 minutes. Add the onions, bell peppers, and celery. Sauté for about 5 minutes, or until the vegetables are wilted. Add the corn bread and 1 cup of the stock and mix well. Season with the salt and cayenne. Cook, stirring, for 2 to 3 minutes. If the dressing seems too dry, add more of the stock as needed for a moist dressing. Remove from the heat.
Serve warm or let cool to room temperature if using as a stuffing.