Turn off antialiasing when drawing things. That's about the only way. Other than that it's merely a matter of technique and you have to work on that, including reading up on how to invert, combine and whatever selections. In fact a reasonably experienced user probably wouldn't bother with the paint bucket at all - you select whatever you need, expand it by whatever amount is needed, fill it on a separate layer and then arrange the layer stack to give whatever visual result you are after.
turning on AA minimizes the non-filled line, but does not completely eliminate, so where it is important, you should get into the habit of using layers. That is, after all, one of the major points of using Photoshop. I've turned the background layer off for the sake of clarity, but you don't need to.
Use the Magic Wand, and make sure Contiguous is checked, and Sample all layers is un-checked.
Select outside the brushed ring.
Invert the selection (Shift Ctrl i) and make a new layer beneath the brushed ring, and fill that with the paint bucket, or better still Alt back space
In actual fact, you'd still run into AA artefacts because it would pill outside the black ring (I have turned on the background and made it black to highlight this.
So to avoid that, use Select > Modify > Contract and reduce the selection by a few pixels.
That is a bit long winded, so what we'd be more likely to do, is let it over spill, and use Free Transform to make it a bit smaller
The point here is that using layers gives you a huge range of options, and it is good practice, and it will make you a better more capable Photoshop user. Honest injun
Just tried it. Thank you.