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.