I feel like it just wasn't a thing for units to come back and fight because I remember reading that people who broke and routed often threw down their swords and shields and anything else they could so it didn't weigh them down when they fled.
In Classical Greece, the rout was usually final, but not always. There are a couple of examples of units rallying and going back into the fray. Take for instance the battle of Solygeia (425 BC), an Athenian amphibious assault on Corinthian territory:
The right wing of the Athenians and Carystians, who had been placed at the end of the line, received and with some difficulty repulsed the Corinthians, who thereupon retreated to a wall upon the rising ground behind, and throwing down the stones upon them, came on again singing the paean, and being received by the Athenians, were again engaged at close quarters. At this moment a Corinthian unit came to the relief of the left wing, and they routed and pursued the Athenians right to the sea, whence they were in their turn driven back by the Athenians and Carystians from the ships.
-- Thucydides 4.43.3-4
The key thing here is that the Corinthians withdrew to a strong, partly fortified position, which renewed their resolve; the Athenians, meanwhile, were driven into the surf and could retreat no further. If either side had had their backs to an open field, it is unlikely that they would have rallied.
Admittedly, fleeing troops often gathered themselves to some extent once they reached their camp or a friendly city wall, which was sometimes a nasty surprise for their pursuers. But this rarely meant that they would rejoin the battle proper.