Gilgal mentioned in this story does not seem to correspond with other places with the same name mentioned in the Tanakh. It appears to be North of Beit El. In the Elisha stories we find that Elisha is a travelling prophet like Shmuel, however, Gilgal is his permanent dwelling. Eliyahu arrives in Gilgal in order to depart from his student.

Courtesy of the Virtual Beit Midrash, Yeshivat Har Etzion