After the first day, slice up the rest of the meat and remove all the meat from the bones, and lay it in a casserole dish or sturdy container. Pour a good quality chicken stock over the meat, to keep the meat from drying out. I suggest Kitchen Basics brand or a similar brand that isn't full of salt, sugar, and preservatives. Cover the dish or container securely.
Many websites indicate that leaving the chicken on the bone makes it dry out quicker, but I don't know if there's any real science behind this. But removing the meat completely does help to cool it down more quickly, and enables you to cover each piece in stock in the fridge. And many people swear that removing the chicken from the bones keeps it from drying, so it's worth trying.
Sauces will only do so much. Leftover rotisserie chicken is great in a soup, in tacos, and in casseroles. You could use chicken stock, noodles, sauteed carrots and celery, and some chicken to make a really fast soup. Shred it and serve in heated taco shells with salsa and cheese and some fresh tomatoes and lettuce. Casseroles with chopped cooked broccoli, cheese, rice and leftover chicken can be delicious and really quick to throw together.