I have seen in translations this being translated as "head covered"
From the perspective that it does not mean "head covered" what would the next best translation be?
What is the best translation of κατὰ κεφαλῆς in 1 Cor 11:4?
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What is the best translation of κατὰ κεφαλῆς in 1 Cor 11:4?
The best translation of κατὰ κεφαλῆς would be (covering-)over the head.
First, we see the exact same Greek prepositional phrase occurs in the Septuagint in the following verse.
Esther 6:12 (LXX)
12ἐπέστρεψεν δὲ ὁ Μαρδοχαῖος εἰς τὴν αὐλήν, Αμαν δὲ ὑπέστρεψεν εἰς τὰ ἴδια λυπούμενος κατὰ κεφαλῆς.
Esther 6:12 (NASB)
12 Then Mordecai returned to the king’s gate. But Haman hurried home, mourning, with his head covered.
In the context, Haman was "mourning," and so he covered his head with something.
The prepositional phrase also occurs in the Textus Receptus version of the Greek New Testament. Since the KJV and Luther Bible translators used the TR, the following verses provide more perspective.
Mark 14:3 (ScrTR)
3 Καὶ ὄντος αὐτοῦ ἐν Βηθανίᾳ ἐν τῇ οἰκίᾳ Σίμωνος τοῦ λεπροῦ κατακειμένου αὐτοῦ ἦλθεν γυνὴ ἔχουσα ἀλάβαστρον μύρου νάρδου πιστικῆς πολυτελοῦς καὶ συντρίψασα τὸ ἀλάβαστρον κατέχεεν αὐτοῦ κατὰ τῆς κεφαλῆς.
Mark 14:3 (KJV)
3 And being in Bethany in the house of Simon the leper, as he sat at meat, there came a woman having an alabaster box of ointment of spikenard very precious; and she brake the box, and poured it on his head.
Markus 14,3 (LUTH1545)
3 Und da er zu Bethanien war in Simons, des Aussätzigen, Hause und saß zu Tische, da kam ein Weib, die hatte ein Glas mit ungefälschtem und köstlichem Nardenwasser, und sie zerbrach das Glas und goß es auf sein Haupt.
The German provides nuance that does not appear in English: that is, the preposition auf here takes the accusative (instead of the dative) and so the idea is not so much as on (dative), but onto (accusative).
In conclusion, based on these references and suggested translations, we may translate the phrase κατὰ κεφαλῆς in English as (covering-)over the head.
For a careful examination of the extra-biblical and potential biblical evidence, I recommend A. Philip Brown II, "Chrysostom & Epiphanius: Long Hair Prohibited as Covering in 1 Cor. 11:4, 7," Evangelical Theological Society 15 Nov. 2011: 1-15
Brown argues that the more plausible understanding of this expression is "having [long hair] on his head," with the bracketed text understood. Brown acknowledges that the case isn't irrefutable, but his thorough review of the evidence and his cogent arguments are convincing. Short of some new evidence appearing, Brown should probably have the last word in this debate.
In Greek, κατα κεφαλης εχων means most accurately, "with his head covered." Literally, to the point of becoming not a translation, it means, to "have upon [his] head" (cf. Mark 14:3 as pointed out by another answer: she poured the spikenard "over his head"). This is an instance where κατα means against/over in the sense of 'draped over, on top of,' and where "to have over the head" is not explicit in form, but clear in meaning, i.e. a head covering—an item of clothing of whatever kind with said purpose of cover the head. So it was understood by the Greek fathers also.
Historically the Greek, κατα κεφαλης εχων translates as "with his head covered" and this happens because of assumptions causing confusion that get straighten out.
The First Assumption
Since κεφαλης literally means "head". Physically minded folks think this references the literal object we call the head. However within the Spirit we can see how literal objects can carry metaphorical consideration. One example of this we locate in Mark 8:15 "yeast of the Pharisees". And just how literal yeast was not the point of the teaching neither is the literal head in this teaching.
Also if we where taking it literally as covering the head as dishonorable, why would the High Priest Aaron receive details for his head covering display? But since acting as High Priest was honorable, covering the literal head is honorable to. Therefore "head" here is used metaphorically.
The Second Assumption
Since in the following context the idea of covering the head is involved the concept "have upon [his] head" infers where as the Greek word κατα stretches in sense as meaning over or against giving the concept as "draped over or on top of" and this concept could work if speaking of literal heads however it would be more correct to say "down on [his] head" even thought the word "his" is totally fabricated for holding this attempt of translation and here is why:
2596 katá (a preposition, governing two grammatical cases) – properly, "down from, i.e. from a higher to a lower plane, with special reference to the terminus (end-point)" (J. Thayer).
We can see here that this word is in the preposition form. And preposition express relation of nouns, pronouns, or phrases with elements in the clause usually defining location. Against works as translation possibility because The A(not) gaining compares well with the concept down.
If the Greek writer wanted the preposition "upon" there is ἐπὶ and ἀνά as options, however they're antonyms of κατα.
The Third Assumption
Since Christians deep within recognize the metaphorical usage in Biblical teachings. The concept of "Authority" often attaches metaphorically with the the concept "headship" however the translation later on quickly cancels the metaphorical attachment with authority and for some reason people return association with head as literal again, but we learned earlier why it is metaphorical.
Well in the process of elimination we can cross out "Authority" as the metaphorical meaning of "head". And eight years later the biggest Valentines Day complaint I've heard from many women when considering their husband made the connection.
"He should already know what I want for Valentines Day, and he didn't get it. I hinted and hinted and even had my girlfriend hinting."
That's right ladies I was listening.
The Root of the Complaint
Let us glorify the wives out their for what they do for their husbands benifit, some make lunches, do laundry, cook, clean, plan, organize, some do it all, and countless other things that could fill up the internet. And why do they do this? Because they are thinking of their man.
What do the men do? We got that nifty spatial relation gift from God, they step out of the way, they wait as they walk past, they zoom through that light, they throw balls, they move stuff. And why do they do this? Because they are thinking of what is happening right now and hopefully thinking of the teachings of Jesus.
Valentines day is that one day expected of man for thinking of their wives. And the offense comes since the woman thinks of the man, they think man should think of the wife. However this teaching in 1 Cor 11 should clear that up and cancel the offense.
2 But I commend y'all because everything of mine you remembered and just as I taught y'all, you are keeping the teachings. 3 But I want y'all to realize because of every man the Template is the leading concern but the leading concern of the wife is the husband but the leading concern of the Template is Aloha. 4 Every man asking or planning down concerns of the Template has shamed himself. 5 But every woman asking or planning not exposing the husband's concerns, shames herself; for it's one and the same as if she shaved her hair off. 6 Indeed if the wife buzzes off her hair by not exposing the husband's concerns, and if disgraceful women shave or buzz their hair, expose the husband's concerns. 7 Indeed man truly ought not expose the head icon and glory of Aloha as under-authority but the glory of man is his woman. 8 For it is not man from out of wife, but woman from out of man, 9 and indeed man was not created through the wife but woman through the man. 10 Through this, out of possibleness, ought through the messages keep upon the woman's head. 11 However, neither woman separate from man, neither man separate from woman in Aloha. 12 Just as indeed in this way the woman from out of the man, and the man through the woman, but everything from out of Aloha. 13 In them y'all judge: Is it proper wife exposing not the concerns of the husband for the Aloha when asking? 14 Doesn't nature itself even teach y'all because if man has long hair it is his dishonor. 15 But if the woman has long hair it is her glory because the long hair is given her instead of the hijab. 16 But if anyone being contentiously thinking, not such our custom, nor we keep in the churches of Aloha.
The "concerns of the Template" is the head of every Christian man. Any planning of bringing down the teachings is dishonorable, but planing on upholding the teachings that is honorable.
1 Cor 11:4 including κατὰ κεφαλῆς should translate as:
4 Every man asking or planning down concerns of the Template has shamed himself.