Adding hay is good it will help achive a better C:N balance. Adding bedding is good too, although I find shavings take longer to compost than pellets. Pine pellets compost great, but I’ve made my best compost with straw pellets.
FYI, earthworms are not impacted by chemicals used to kill internal paritsites which are not actually “worms.”
If you are planning to share your compost expect a landscape company to ask about the herbicides used on your hay and pastures. There are a few persistent broad leaf herbicides that are not broken down in the the process that remain active in compost for a year or so. The affected compost can damage tender plants. Google “Persistent herbicide in compost” for more information. I have not had any difficulty finding out from my hay sources what herbicides were used on it. (If you feed hay that contains alfalfa or clover that would not have been treated.)