You're right
You're right, Zigûr. While I don't have it to hand, I recall in the relevant essay in Unfinished Tales that the Éothéod moved to the upper vales of Anduin after the fall of Agmar, to seek more room.
By the time of Eorl, they had become too many for that location, but were constrained from moving south because of the growing power of Dol Guldur, occupied by Sauron. Therefore, the offer by Ruling Steward Cirion of Calenardhon to Eorl as a reward for their help would give them plenty of room to grow.
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