Paul did address head coverings
The main passage is Paul’s letter to the church in Corinth. He speaks about what men and women should do with their heads when they pray or speak in the gathered church. Men were not to pray with their heads covered, while women were to have a covering rather than pray with their heads uncovered (1 Corinthians 11:4-7).
This is more specific than a general rule about any hat someone might wear on a Sunday. Paul was dealing with visible practices during worship, and he tied them to honouring God and respecting the relationships he had just described between Christ, man, and woman (1 Corinthians 11:3).
But I would have you know, that the head of every man is Christ; and the head of the woman is the man; and the head of Christ is God.
Why does the passage feel difficult?
Corinth was a busy Greek city with strong social expectations about dress, status, sexuality, and public respectability. A head covering could communicate things to people in that setting that it may not communicate in the same way now. Paul wanted the church’s worship to point towards God’s good order, rather than cause needless scandal or confusion.
He also appeals to what the Corinthians themselves recognised as fitting, then mentions hair in his discussion of a woman’s covering (1 Corinthians 11:13-16). That has led faithful Christians to different conclusions. Some understand Paul to require a cloth covering for women whenever the church gathers. Others believe the enduring principle is modest, honouring worship, while the particular form of covering was shaped by Corinthian culture.
Judge in yourselves: is it comely that a woman pray unto God uncovered? Doth not even nature itself teach you, that, if a man have long hair, it is a shame unto him? But if a woman have long hair, it is a glory to her: for her hair is given her for a covering.
What about hats today?
A hat today may be worn for warmth, sun protection, cultural custom, celebration, or simple personal style. Its meaning depends greatly on the setting. The Bible does not directly discuss baseball caps, fascinators, beanies, or formal hats as modern items.
If a man chooses to remove his hat during prayer or a church service because he believes it expresses reverence, that fits naturally with Paul’s instruction to Corinthian men (1 Corinthians 11:4). If a woman wears a hat or head covering as an expression of faith and honour, she should be free to do so sincerely. Neither practice should become a reason to look down on others.
Choose reverence, not judgement
Paul’s concern was not fashion policing. He wanted believers to honour Christ and to consider how their conduct affected the worshipping community. He describes a woman’s covering as connected with honour and appropriate conduct in prayer (1 Corinthians 11:5-6, 1 Corinthians 11:10).
So if your church has a custom around hats or head coverings, learn its reasons and respond with humility. If it does not, don’t assume someone is less spiritual because they wear a hat, remove one, or cover their head. Read the passage for yourself, ask how it points you towards honouring Christ, and worship with a heart turned towards God.