[Moving Selections question]

Is there any way to move a selection without replacing other contents of that layer?

[Example] I try to move the selection to the left, but it covers the cheek to the left:

Here's the same thing in MS Paint, but with the desired result:

There is no white in the current layer. Only the black lines.

I do know that it's possible by making a new layer, but I'd rather keep my lineart to one.