There is a discussion about this in Preston T. Massey, "The Meaning of κατακαλύπτω and κατὰ κεφαλῆς ἔχων in 1 Corinthians 11.2–16,"
NTS 53 (2007): 502-523. For what is worth. here is the relevant paragraph and footnotes:
Massey 2007:522 wrote:At this point a final conclusion may be drawn with regard to the expression κατὰ κεφαλῆς ἔχων as used in 1 Cor 11.4. This idiomatic phrase does not mean ‘letting down the hair’. This expression, though generally used with an object (but understood without the object), always implies some kind of garment or cloth coming down from the head.[71] Furthermore, κατὰ κεφαλῆς ἔχων approximates the nearly identical expression in 1 Cor 11.10 of ἔχειν ἐπὶ τῆς κεφαλῆς, which also employs terminology characteristic of a head covering.[72] This expression is normally used with the familiar textile garment himation.[73] The himation is a synonym for the περιβόλαιον mentioned at v. 15. Therefore, the text is talking about veils or material head coverings, a subject perfectly familiar to native Greeks.
71. Besides the references already cited, κατὰ τὴς κεφαλῆς τὸ ἱμάτιον (a covering down from the head) may be found in the following texts: Dionysius of Halicarnassus The Roman Antiquities III.71, VI.3.3, XI.26.4, XII.16.4, XV.9.7, XIX.8.3; Plutarch Pompey XL.4; Caesar LXVI.12 The Sayings of the Romans 200F; The Roman Questions 266C and 267C; Fortune of the Romans 319C (which has the similar ἀφεῖλεν ἀπὸ τῆς κεφαλῆς τὸ ἱμάτιον; and Josephus, Ant. III.270. See also in similar but not identical language Josephus J.W. VII.128 and Plutarch’s Pompey VIII.3. These texts all describe a veil or head covering. Going back further to archaic times, the Hymn to Demeter (180) contains the line στεῖχε κατὰ κρῆθεν κεκαλυμμένη.
72. Diodorus Siculus Histories I.62, uses the exact phrase ἔχειν ὲπὶ τῆς κεφαλῆς (‘to wear upon the head’). The object worn upon the head in this instance is the head of an animal. The use of the infinitive ἔχειν combined with κεφαλή in the genitive case describes some artificial object placed upon the head; it does not describe long hair hanging down. The very same expression is also found in Pausanias Description of Greece (Achaia) VII.5.9 and (Arcadia) VIII.32.1. Plutarch uses a participial phrase in The Roman Questions 266C: τύχωσιν ἐπὶ τῆς κεφαλῆς ἔχοντες τὸ ἱμάτιον. The use of the verb ἔχειν as conveying a temporary condition is perhaps intended as a contrast to a more permanent condition as reflected in the verb εἶναι (as in Pausanias Description of Greece [Phocis] X.30.8).
73. Such as Pausanias Description of Greece (Laconia) III.15.11, who describes Aphrodite as κάθηται δὲ καλύπτραν τε ἔχουσα (‘who sits wearing a veil’). There is one very anomalous construction in Josephus: τὴν κόμην. . .ἔχων (Ant. VII.267). This rare reference, however, is not associated with the adjective ἀκατακάλυπτος. Josephus, in accordance with convention, uses the appropriate verb ἀνίηνι (V.347) and the noun ἂνέσις (V.285) to describe loosened
hair.
In my view, there is an implicit object with ἔχειν, and supplying that object, whether a coverning or hair, requires cultural knowledge we may not have access to.
Stephen C. Carlson, Ph.D.
Melbourne, Australia