The chlamys was a short riding cloak worn by men in ancient Greece, especially among cavalry men and ephebes. It was believed to have originated among the horsemen of Thessaly.

As the following quote from Plutarch’s life of Philopoemen shows, the chalmys eventually became standard attire, even for the general soldiery.

Then, while the minstrels were contending for the prize, he came into the theatre with his young men. They wore their soldiers’ cloaks and their purple tunics, were all in the prime of their strength and of the same age, and showed not only great respect for their commander, but also that high spirit which young men have after many honourable contests.

Plut. Phil. 11

A petasos is a broad-brimmed travelers hat of Thessaly.

χλα^μύς [υ^], ύδος, ἡ: acc. χλαμύδα, also
*A. [select] “χλάμυν” Sapph.674:—short mantle, worn prop. by horsemen, X.An.7.4.4; borrowed with the πέτασος from Thessaly, Philem.34, Poll.10.124; but said to be Macedonian, Arist.Fr.500, Phylarch.62J.; worn by ἔφηβοι, Philem. l.c., cf. AP6.282 (Theod.); μάτηρ σε . . δῶρον ἐς Ἅιδαν ὀκτωκαιδεκέταν ἐστόλισεν χλαμύδι ib.7.468 (Mel.); χλαμύδεσσ᾽ ἀμφεμμένοι, of ephebi, IGRom.4.360.35 (Pergam., ii A. D.); ἐκ χλαμύδος, = ἐξ ἐφήβου, Plu.2.752f, cf. 754f; “ἐκ χλαμύδος . . ᾤχετ᾽ ἐς Ἅιδα” IG12(7).447.6 (Amorgos); worn by Hermes, Luc.Tim.30; also by Eros, Sapph. l.c. (v. Poll.10.124), Philostr.Im.1.6, cf. AP12.78 (Mel.).
2. [select] generally, military cloak, of foot-soldiers, Antiph.16, Men.331, Plu.Phil.11, etc.; of heralds, Ar.Lys.987.
3. [select] of the general’s cloak, Phld.Vit.p.27J., Plu.Per.35, Lys.13, etc.; worn by kings, Id.Demetr.42, etc.; by tragic kings and heroes, Luc.JTr.41; by Σειληνοί in a procession, Callix.2: = Lat. paludamentum, D.C.59.17, 60.17, al., Hdn.4.7.3, Cod.Theod.14.10.1.
4. [select] a civilian’s mantle, PCair.Zen.263.2, al. (iii B. C.), PLond.2.402 ii 16 (ii B. C.), X.Eph.1.8 cod., POxy.1288.24 (iv A. D.). (For its shape cf. Plu. Alex.26.)

πέτα^σ-ος , ὁ, also ἡ Eratosth. ap. Ath.11.499e ; πετάσῳ Θετταλικῇ is prob. cj. for πίλῳ Θετταλικῇ in Thphr.HP4.8.7 (cf. 9) : (πετάννυμι) :—
A.broad-brimmed felt hat, worn by ἔφηβοι and hence used as their badge, Poll.10.164, Suid.; γυμνάσιον καθίδρυσε καὶ τοὺς κρατίστους τῶν ἐφήβων ὑπὸ πέτασον ἦγεν, i.e. made them practise gymnastics, LXX 2 Ma.4.12 ; also in representations of Hermes, Ephipp. ap.Ath.12.537f.
II. from its shape, broad umbellated leaf, as of the lotus, Thphr.HP4.8.9 ; φύλλον μέγα ὡς π. Dsc.2.106.
III. from its shape, also, awning, “ὁ π. τοῦ θεάτρου” OGI510.4 (Ephesus, ii A. D.), CIG3422.17 (Philadelphia) ; also, of the circular tomb of Porsenna, Plin.HN36.92; baldacchino, PMag.Leid.W.3.11.

https://en.wikipedia.org/wiki/Chlamys

https://en.wikipedia.org/wiki/Petasos