Interesting. How do you know this to be true?
"True" is a strange ask when we started with a mythological story. But it does seem to have some factual basis. The story comes from the Edomites - hence their eponymous ancestor is named Adam/Edom.
The southern area is easily verified. The Bible names the cities in that region, tells us they were living there at the time of writing, and we have an archaeological record that matches.
The northern area and the trek southwards is also in the Bible, but it's harder to see. There was no such thing as a written history in the Bronze Age, but they
were concerned with recording alliances between tribes. In the Bronze Age, when a marriage is recorded, that's meant to record an alliance.
In Genesis, Edom is recorded as taking 4 wives. The first two are Hittites. The southernmost extent of the Hittite empire is almost the top of the map I posted.
The 3rd wife/alliance is a Hivite. The Hivites are recorded to have lived in the valley at the base of Mt Hermon, which is just about the center of the top circle on the map (that's why the circle is drawn there). But the Bible also records that an entire tribe of Hivites left their homeland and went with the Edomites into the wilderness.
The 4th was an Ishmaelite. The Ishmaelites were nomads who lived close to the southern circle. Further, the locations of the birth of Esau's grandchildren is given. The first ones are born "in the land of the Canaanites" (aka the north) while the later ones are born "in Mt Seir" (the south).
Gen 36:8 Thus dwelt Esau in mount Seir: Esau is Edom.
The Bible also contains the same story a couple more times.
Perhaps you recall the story in which Edom sold/lost his birthright to his brother Jacob and had to leave the area? And later there's a story where the two grown brothers with their tribes meet each other going opposite directions.
Historically, what seems to have happened is the Edomites made alliances with the losers of a war for that desirable northern territory and had to vacate.