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?
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.
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.
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.