Add 6 cups water, onion and celery trimmings, and shrimp peels to large pot. Bring to a boil. Lower heat and simmer for 20-40 minutes. Strain the broth and measure it; you will need 4 cups (or double the amount of rice), so add water if necessary.
Heat the oil in a 7- to 8-quart enameled cast-iron Dutch oven (or other heavy-duty pot or high sided skillet) over medium high heat and add the andouille, onion, celery, bell pepper, and Creole seasoning. Cook, stirring often, until the vegetables are browned, 15 to 20 minutes. Add the ham near the end of the cooking time so that it doesn’t overcook.
Add the rice and cook, stirring often, until translucent, 2 to 3 minutes, to toast the rice slightly.
Add the reserved broth, garlic, thyme, bay leaf, and 3/4 tsp. salt; bring to a boil. Cover, lower the heat to maintain a simmer, and cook, undisturbed, just until the rice is tender, about 20 minutes.
Add the shrimp on top of the rice with about 5 minutes left to cook, covered.
Remove the pot from the heat. Using a fork, gently fluff the scallions into the jambalaya. Let stand for 10 minutes to let the flavors meld.