As the Lord thy God liveth, there is no nation or kingdom, whither my lord hath not sent to seek thee: and when they said, He is not there; he took an oath of the kingdom and nation, that they found thee not. (1 Kings 18:10)
There were many small kingdoms not far distant from Israel. It would be natural for someone whose life was hunted to seek exile in some nearby state. Ahab not only searched his own land but had inquiry made for Elijah in all the neighboring countries.