There is no contradiction at all.
Paul specifies, quite clearly :
Every woman that prayeth or prophesieth with her head uncovered dishonoured her head [1 Corinthians 11: 5.
The woman referred to, above, was performing a domestic service - the washing of feet - which had been neglected to be organised by the host of the gathering. So she performed the service, using all she had with her to so perform it. Not having a basin of water or a towel at her disposal, she used her own tears and her own hair.
She was neither ‘praying’ nor ‘prophesying’.
It is highly likely, almost certain, in those days and in those parts, that, in any case, she would have had her head covered whilst in public and whilst in mixed company. Having long hair, as all women in that time and in that place had, she would have quite easily performed the service without removing her head ‘down-veiling’ (as is the original in 1 Corinthians 15) as the long hair would have protruded from beneath the head covering.
What Paul specifies is an ordinance in the church, witness his words :
If any man seem to be contentious, we have no such custom [1 Corinthians 15:16, KJV],
Meaning that they had no such custom in the churches as to leaving the head uncovered.
It is also abundantly clear that ‘hair’ is not the ‘covering’ Paul is referring to.
He says that if a woman be not ‘covered’ let her ‘also’ be shorn, indicating that being without a ‘covering’ still leaves her with hair which could be shorn off.
The reasons Paul gives for the ordinance are :
- The Headship of Christ
- the headship of the male
- the image and glory of God
- the glory of the man
- the creation of man
- the creation of woman
- because of the angels
- the teaching of nature
- the custom of the churches
None of these twelve reasons is anything to do with Jewish custom or with peculiar customs of the times. They are all to do with creation and the order of creation while the church waits for full and final redemption at the coming of the Lord Jesus Christ.
Every single reason he gives is as relevant today as the day Paul dictated the epistle and signed a salutation with his own hand on the parchment.
In my childhood, sixty years ago, in the Highlands of Scotland among both Presbyterians, Episcopalians, and Evangelicals, it was unthinkable that a woman should attend a church meeting without a head covering.
The modern practice, which seems to me to have gained ground in the early 1960s, and is now evident in almost all self-professed Christian denominations, is an innovation, unheard of in former times.