Heat oil in a Dutch oven over medium-high heat. Add chicken and saute until no longer pink and juices run clear, 5 to 7 minutes. Season with Cajun seasoning. Set aside in a separate dish.
Cook and stir sausage in the same pot over medium-high heat until nicely browned, 5 to 7 minutes. Add ham and saute 2 to 3 minutes more. Transfer mixture to the chicken dish.
Heat remaining grease and olive oil as needed in the same pot over medium-high heat. Add onion, celery, bell pepper, and garlic; saute until onion becomes translucent, about 5 minutes. Stir in remaining Cajun seasoning, paprika, oregano, basil, and thyme.
Pour in chicken broth and stir to mix. Add meat mixture. Add tomatoes, tomato soup, bay leaves, and hot sauce; mix thoroughly. Add rice slowly and bring to a boil. Reduce heat to a simmer and let cook, stirring occasionally to prevent sticking, until rice is tender, 45 minutes to 1 hour.
This is fairly spicy so if less heat is desired substitute paprika for the Cajun seasoning and omit hot sauce, then add salt and pepper to taste.
You can just remove the sausage casings instead of dicing, if preferred.
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