The Apostle Paul wrote the passage about not requiring to veil in 1 Corinthians 11:1-16,
He progresses through a discourse to the Corinthians for which he provides no historical context as to whether the Corinthians were encountering some problem for him to bring it up, yet He culiminates the passage very clearly.
Let's step through some of the culmination.
VERSE 12:
Paul wrote "For as the woman originates from the man, so also the man [has his birth] through the woman; and all things originate from God" (1 Corinthians 11:12).
Paul is narrowing the focus to God. His enumeration in verse 12 culiminates with "all things originate from God".
VERSE 15:
Paul wrote "if a woman has long hair, it is a glory to her. For her hair is given to her for a covering" (1 Corinthians 11:15).
A glory is not dishonor. Paul mentions dishonor/disgrace for a woman to pray uncovered in verses five and six.
Paul is writing about a woman's head covering, swithing between a veil/garment/clothing and hair throughout the passage of 1 Corinthians 11:1-16; however, he is not separating purpose.
Notice that Paul wrote "the man is the head of a woman" in verse 2, and "every woman who has her head uncovered while praying or prophesying dishonors her head" in verse 3.
Paul integrates this dishonor versus honor concept and the glory concept as a single thing; however, the original poster illegitimately attempts to separate "submission" and "glory".
"Her hair is given to her for a covering"
This clause is HUGE. The English "for" in the clause is translated from the Greek "anti". The NASB and ESV and KJV translations use the word "for". The Greek anti means "in place of" thus the English "for" functions accurately in 1 Corinthians 11:15.
For the woman, her long hair is a glory to her.
VERSE 16:
Paul wrote "we have no such custom, neither the assemblies of God" (1 Corinthians 11:16).
These are the concluding words of the passage - the passage presented by the original poster.
These are the culminating words of the passage.
"we have no such custom" is written about women's head coverings, and the verse culiminates the passage. It is a precise statement by Paul.
A woman need not put a garment/clothing/veil on her head to pray, so says the Apostle Paul.