If your used moving boxes are still in good shape and you’d like them to be put to good use instead of recycling them, you might consider giving them to Goodwill, who could always use them to transport items or use them to store new donations.
If you are considering this option, try getting in touch with your local Goodwill or other charity location, like the Salvation Army, and ask them if they could use any empty boxes. The chances are good that they will be happy to take your empty moving boxes off your hands.