Bring converted rice and 3 cups water to a boil in a saucepan. Reduce heat to medium-low, cover, and simmer until the rice is tender and the liquid has been absorbed, 20 to 25 minutes.
Heat a large soup pot over medium-high heat. Cook and stir beef in the hot pan until browned and crumbly, 5 to 7 minutes; drain and discard grease.
Stir 2 cups water, cooked rice, pasta sauce, tomato soup, cabbage, onion, salt, black pepper, and hot sauce into beef. Bring soup to a boil, reduce heat to medium-low, and cover. Simmer until cabbage is soft and tender, 20 to 30 minutes.