Yes, you can add the cooked rice to the whole soup recipe but my personal preference is to add the rice to each individual serving so it doesn't soak up too much of the broth. If you know you'll be serving the entire recipe at once, it's okay to go ahead and add it. But if you are making this soup for meal prep or plan on having leftovers, I think it's best to add the rice as you serve it. If the rice sits in the broth overnight, it will start to soak it up and can become mushy and make the soup too thick.