Hobbits don't migrate to the Shire until the middle of the Third Age, thousands of years after this series, and for good reason, since the area is occupied for quite some time by the heirs of Elendil ruling the kingdom of Arnor until the efforts of the Nazgul bring it down. Aragorn was of the lineage of the kings of Arnor, and had claim to Gondor because Arnor had seniority as a kingdom over Gondor, as the latter was founded by Elendil's sons.
It is possible that the Harfoots could do something to earn title to land during this series though. Prior to migrating west, Tolkien's works hint that they lived in the river valley of the Anduin as this is where Smeagol/Gollum lived in his youth. This region, which corresponds to the territory the Fellowship travels through on the Anduin after leaving Lorien, lies to the north of Gondor and is hypothetically sufficiently within the writ of Isildur that he could award it to them following the victory of the Last Alliance.