In large saucepan over medium heat, heat 1 tablespoon oil. Add onion, green pepper and celery. Cook 5 minutes or until vegetables have begun to soften. Add garlic. Cook 2 minutes. Remove from pan. Reserve.
In same saucepan over medium heat add remaining ¼ cup oil and flour. Cook 12 to 15 minutes, stirring continuously until mixture is brown and fragrant. Whisk in broth. Bring to boil to thicken. Reduce to simmer.
Add reserved vegetables, shredded turkey and gravy, andouille, Worcestershire sauce, Cajun seasoning, hot sauce, cayenne, thyme and bay leaves. Season with salt and pepper, to taste.
Cover. Simmer, stirring occasionally 15 to 20 minutes. Discard thyme sprigs and bay leaves. To serve, ladle gumbo over rice.