What term means "the area beyond the surrounding area"?

Solution 1:

Outskirts is the correct term I believe.

John lived in the outskirts of the city

Outlaying areas is also correct, but certainly less common and seems a little less conversational.

Solution 2:

I visualize a circle with a person at the center. The horizon represents a belt around the circumference. What's beyond that belt is what we're trying to name.

The previously suggested word "outlying" could work, but it doesn't definitely imply the space outside of that belt we're calling something like "the surroundings". If you're trying to maintain a level of certainty with this term, and you already have a sense and even a name for the inner territory and the surrounding territory, I would suggest some compound using "extra-".

"extra-" a prefix meaning “outside,” “beyond,” freely used as an English formative.

Perhaps some form of extra- + [adjective form of name for area inside the circle]:

• Extra-territorial (if you're talking about an expanse of land)
• Extra-spacial (if you're talking about dimensions)