Throwing out bread

After shabbat, we often have lots of pieces of bread left over, which experience shows will never get eaten; rather, they just sit in the bread box until they’re so stale that they’re inedible. Can we throw these out right away, or do we have to wait til they become rock hard and disgusting?

You can throw out right away.

I would like to add to this discussion. My grandmother z"l lived through two world wars. I remember a picture of her from around 45/46, where she weighed about 45kg. She never threw away food, and I learned from her.
We never throw away any bread. I put it in a paper bag and leave it there to dry out. When the time comes and I have enough dry bread, I put them in a food processor and make breadcrumbs out of them. This will make a mix of crumbs, as I use not just challah, but also wheat and rye breads or any kind of bread that gets old ad dry. I have not bought store made breadcrumbs for the last 30 years.

Beautiful and commendable but halachicly problematic. Make sure you wash for and eat bread on all those meals when you serve these breadcrumbs. Commercially made breadcrumbs are intended and made to be ground up and that intention nullified the halachic status of bread. Whereas your breadcrumbs were actually bread and have the halachic status of bread and grinding them up does not remove that status.