Learn English – What term means “the area beyond the surrounding area”

I think this is best described by "evolution." It can be used in whatever form works in context. For example, "The town had evolved into a place he no longer recognized."

The word twilight actually means what you're looking for:

twilight, n.: the light from the sky between full night and sunrise or between sunset and full night

Animals that are active during twilight are said to be crepuscular, so if you're looking for an adjective to apply to that period, crepuscular (which is itself derived from the Latin for twilight) should serve.

Edit: Lacking the context in which you wish to use the term you're looking for, I can offer these suggestions:

Sunset/sunrise occur at the instant that the sun's upper limb is tangent to the horizon. You could then, for example, have science officer of some enterprising starship reporting, "Two degrees to limb tangency, Captain" to mark some impending sunrise/sunset/moonrise/moonset/et ceteraset.

Sunset/sunrise occur at a precise zenith distance of 90.8333 degrees. You could coin a term for this value, so that the same science officer could report, "Two arc seconds from zenith distance alpha, Captain."