I don't think of them as bad unless they are leaf cutters and eating my food (or biting me). When you start digging they are going to move along and build new tunnels. They're generally moving around a lot anyway. If you want to encourage them to start moving away from where you're going to be digging, you could start disturbing that hole right now and they'll get the point. Or put a piece of wood on top of that entrance (under your bucket). I know there are many kinds of ants and probably not every one is the same but it is very likely that they have various entrances and you're not going to doom the whole colony by closing off one entrance.
Are ants bad? If so, what's the most efficient, non toxic way to eliminate them? If not, is there a way to encourage them to move somewhere better?
Ants are a part of the soil food web and play important roles in nature. Often, disturbing an ant hill by kicking it with your foot or pushing it over with a shovel will cause the ants to abandon that spot and relocate.
"Study books and observe nature; if they do not agree, throw away the books." ~ William A. Albrecht
I try not to bother ants. As James mentioned they are an important part of soil building & the food web. They (or any insect) can be a serious problem if they infest an electrical box though. I remember a post by Dr. Redhawk saying a good way to eliminate them is to spread fresh coffee grounds 3 feet around a mound & water it in. Apparently that will kill the queen.
Argue for your limitations and they are yours forever.