Greek and Roman literature is certainly an important collection of evidence for clothing in the Achaemenid empire. Most of these passages describe the clothing of the king and satraps, or simply say that such-and-such is the Persian equivalent of a Greek garment. Herodotus and Strabo provide information about the garments of other people. Herodotus says that Babylonian men dress as follows:
Hdt. 1.195: For clothing, they wear a linen tunic (kithōn), reaching to the feet; over this the Babylonian puts on another, woolen, tunic (kithōn), and wraps himself in a little white mantle (chlanidion leukon): he wears the sandals (hupodēmata) of his country, which are like Boeotian sandals. Their hair is worn long, and bound with a girdle (mitra); the whole body is perfumed. Every man has a seal and a carved staff, and on every staff is some image, such as that of an apple or a rose or a lily or an eagle: no one carries a staff without a device.
τὰ μὲν δὴ πλοῖα αὐτοῖσι ἐστὶ τοιαῦτα: ἐσθῆτι δὲ τοιῇδε χρέωνται, κιθῶνι ποδηνεκέι λινέῳ, καὶ ἐπὶ τοῦτον ἄλλον εἰρίνεον κιθῶνα ἐπενδύνει καὶ χλανίδιον λευκὸν περιβαλλόμενος, ὑποδήματα ἔχων ἐπιχώρια, παραπλήσια τῇσι Βοιωτίῃσι ἐμβάσι. κομῶντες δὲ τὰς κεφαλὰς μίτρῃσι ἀναδέονται, μεμυρισμένοι πᾶν τὸ σῶμα.  σφρηγῖδα δὲ ἕκαστος ἔχει καὶ σκῆπτρον χειροποίητον: ἐπ᾽ ἑκάστῳ δὲ σκήπτρῳ ἔπεστι πεποιημένον ἢ μῆλον ἢ ῥόδον ἢ κρίνον ἢ αἰετὸς ἢ ἄλλο τι: ἄνευ γὰρ ἐπισήμου οὔ σφι νόμος ἐστὶ ἔχειν σκῆπτρον.https://penelope.uchicago.edu/Thayer/E/Roman/Texts/Herodotus/1D*.html#195
Herodotus never claims to have visited Babylonia like he says he visited Egypt, and he is probably referring to the dress of wealthy and respectable men.
Strabo’s section on Babylon is cribbed from Herodotus, but he has more to say about the Persians of Persis. He seems to draw on a source from the 4th or 3rd century BCE, because his descriptions resemble Xenophon’s Cyropaedia and Diodorus’ account of the wars of the Diadochoi after Alexander’s death.
Strabo, Geography, 15.iii.19 “They are armed with a rhomboidal wicker shield (gerron), and besides quivers they have axes (sagareis) and cleavers (kopidas), and about the head they have towering felt hats (pilēma). Their breastplate is scaled. The dress of the rulers is a triple pair of trousers (anaxyris), a sleeved double tunic (chitōn xeiridōtos) as far as the knees, the undergarment being white (leukos), the overgarment colourful (anthinos). The summer cloak (himation) is purple or violet, the winter colourful. Their tiaras resemble those of the Magi, a deep double shoe (upodēma koilon). Among the masses a double tunic (chitōn) as far as the middle of the shin, and a rag of sindon about the head; each of them has a bow and a sling.
ὁπλίζονται δὲ γέρρῳ ῥομβοειδεῖ, παρὰ δὲ τὰς φαρέτρας σαγάρεις ἔχουσι καὶ κοπίδας, περὶ δὲ τῇ κεφαλῇ πίλημα πυργωτόν, θώραξ δ᾽ ἐστὶν αὐτοῖς φολιδωτός. ἐσθὴς δὲ τοῖς ἡγεμόσι μὲν ἀναξυρὶς τριπλῆ, χιτὼν δὲ χειριδωτὸς διπλοῦς ἕως γόνατος, ὁ ὑπενδύτης μὲν λευκός, ἄνθινος δ᾽ ὁ ἐπάνω: ἱμάτιον δὲ θέρους μὲν πορφυροῦν ἢ ἰάνθινον, χειμῶνος δ᾽ ἄνθινον, τιᾶραι παραπλήσιαι ταῖς τῶν Μάγων, ὑπόδημα κοῖλον διπλοῦν, τοῖς δὲ πολλοῖς χιτὼν ἕως μεσοκνημίου1 διπλοῦς, ῥάκος δὲ σινδόνιόν τι περὶ τῇ κεφαλῇ: ἔχει δ᾽ ἕκαστος τόξον καὶ σφενδόνην.
Strabo’s double tunic seems to be the same as the undergarment and the overgarment. Its just possible that he means a white tunic sewed inside a colourful tunic as a lining. Greek artists don’t seem to have been interested in accurately portraying the clothing of foreigners, and what Greek writers say should be treated with caution too. But its still worth reading their descriptions, because Persians and Babylonians did not leave us disinterested descriptions of their own cultures.
Further Reading: Shapur Shahbazi, “Clothing ii. In the Median and Achaemenid Periods,” Encyclopaedia Iranica https://www.iranicaonline.org/articles/clothing-ii