In the past several years in Pentecost I have met and dialogued with individuals who hold that within 1 Corinthians 11 Paul is noting not only hair, but also a material veil. I do proceed with some bias since I do not believe this doctrine to reflect the meaning of the text of Scripture. Oneness theology should not incorporate such a doctrine into its formal or informal theological profession. There are pastors within the largest Oneness organization (UPCI) in the world, however, who hold to this view (mainly Canada and Jamaica).
First, it is incumbent on one who believes to prove that there is a kalumna or in English veil found in the 1 Corinthian 11 text. The only reference to a material veil would be at the end of the most relevant textual data (1 Corinthians 11:1-16). In vs. 15 the English word "covering" there is a noun, in the Greek "peribolaion" (peri-bol'-ah-yon). Earlier "covering" references are verbs, in all its prior uses in 1 Corinthians 11. Therefore, it specifically says the long hair is what acts as the covering for a woman, in the place of any customary veil wearing.
I see no demand for a second covering, in fact the hair is given naturally and created for the natural order of society. Not a man made material that, if taken literally, no second covering adherent would want to wear in public. Nor could she pray without being covered, in public or private worship, since believers are to pray without ceasing (1 Thessalonians 5:17). The woman’s hair is a natural veil, that is why the reference to a material, man made one, may only be implicit and secondary.
It matters not what other churches or people would say on this matter for this is the Word of God. Paul goes through so much effort to culminate to this point, he appeals to everything possible to make his point (Nature, Angels, Properness), he even reduced an earlier argument to absurdum.
The word glory--doxa--here refers to the grounds, occasion or source of honor or glory, which means her glory, is due her because of her long hair.1 Her hair is "her glory" it is what she should be most concerned with and care for with Godly pride. God's opinion of her is elevated by her obedience to the divine order He has created, not fabric.