When you say "Hellenistic times", do you mean the Hellenistic period (from the death of Alexander the Great to the battle of Actium, 323-31 BC)? Or do you mean the Classical period (from the Persian Wars to the death of Alexander, 490-323 BC), the time of greatest prominence for city-states like Athens and Sparta?

Our evidence for Spartan combat equipment is very sparse for both periods. Even so, it's important to distinguish between the two, because the reforms of Kleomenes III in the 230s BC turned the remaining Classical-style Spartan hoplites into Macedonian-style pikemen. Our only source on this (Plutarch's Life of Kleomenes 11.2) doesn't discuss body armour, and only mentions that the new corps of 4,000 hoplites were armed with pikes wielded in both hands, and that they gave up their big round aspis shields in exchange for smaller Macedonian shields carried by a strap around the neck. Given the trend for pikemen of this period to carry ever heavier armour, it is quite possible that this reform also involved the reintroduction of pieces of protective gear such as chestplates and greaves.

Before this time, as far as we can tell, Spartan warriors were equipped in much the same way as their counterparts in the rest of the Greek world. Light troops wore no armour; cavalry, especially in the Classical period, would be heavily armoured, but would carry open-faced helmet types like the Boiotian helmet and fight without a shield. The hoplites would initially have worn bronze cuirasses, face-covering bronze helmets and greaves as a mark of status. A surviving statue of a Spartan from the early Classical period shows him without body armour, but shows his beautifully decorated Attic-type helmet. However, as the 5th century BC wore on, hoplites throughout the Greek world seem to have shed much of their armour, first switching to glued or quilted linen cuirasses (already common during the Persian Wars) and finally abandoning body armour altogether. By the time of the Peloponnesian War, it seems most hoplites opted for a simple conical helmet called the pilos, and perhaps a pair of greaves, but no other protective equipment except their large shield. It was still possible for leisure-class warriors like the Spartiates to wear bronze armour into battle, but it would have been more and more the exception to do so. Thucydides' account of the battle of Sphakteria suggests that the Spartans deployed there were all wearing pilos helmets, and little else for armour.

What set the Spartans apart was not the quantity or type of their equipment, but its uniformity. By the end of the 5th century BC, all Spartiates and the men they equipped to flesh out their formations were carrying identical bronze-plated shields decorated with the lambda (Λ) for Lakedaimon, the region in which Sparta was situated. They also wore an item of clothing called the phoinikis. The name simply means "the Phoenician thing", referring to the origin of the red dye used to colour it; scholars are in disagreement over whether the phoinikis was a tunic or a cloak. What we do know is that the combination of red clothes and polished bronze shields (and presumably helmets) made advancing Spartan hoplites a daunting sight. Xenophon describes specifically how their army appeared like a single mass, "all bronze, all scarlet" - in contrast to normal Greek armies, which observed little to no rules regarding their equipment, and turned out as they pleased. This description, again, suggests that the Spartans by Xenophon's time (first half of the 4th century BC) wore no cuirasses over their clothes; how else could the red of their uniform intimidate the enemy?

It's worth noting that both the lambda shield and the red phoinikis are first attested at the time of the Peloponnesian War. We have no evidence to suggest that the Spartans already wore these things at the time of the Persian Wars; it seems most likely they didn't, or Herodotos would have remarked upon it. Whether or not the Spartans should wear their distinctive uniform in your story depends on when it is set.