Aligning cut layers in Photoshop

I'm retouching a picture of the eclipse by cutting an egg-shaped section and putting it on its own layer, so I can work with concentric banding on the background. If I select a section and then create a new layer via cut, the two pieces do not align perfectly, and there's a thin white line around one side of the cut section when viewed together.

Did I do something wrong to mis-align the cut sections? Is there an automatic process to re-align them? (The auto-align functions all seem intended to align two similar pictures by matching similar parts in this picture, so they don't apply here.)

I could sort of address the problem by creating a new layer via copy, instead of cut, so the distinction between the two sections isn't so obvious. It would work with this particular picture. But I've had trouble with this in the past when I was inserting a person into a group photo and the perfect alignment of two cut sections of the same image was important, and it seems like there should be a straightforward way to deal with this issue. Any suggestions?

Thanks,

-Rankin