The wearing of the head veil by women at the holy Mass is an ancient tradition.  St. Paul writes of this practice in the eleventh chapter of his first letter to the Corinthians.  

He begins the subject this way,
“Be ye followers of me, as I also am of Christ.  (2) Now I praise you, brethren, that in all things you are mindful of me: and keep my ordinances as I have delivered them to you. 

What are his ordinances?
“(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.  (4) Every man praying or prophesying with his head covered, disgraces his head.  (5) But every woman praying or prophesying with her head not covered, disgraces her head : for it is all one as if she were shaven.  (5) For if a woman be not covered let her be shorn.  (6) But if it be a shame to a woman to be shorn or made bald, let her cover her head.  (7) The man indeed ought not to cover his head because he is the image and glory of God, but the woman is the glory of the man.  (8) For the man is not of the woman, but the woman of the man.  (9) For the man was not created for the woman, but the woman for the man.  (10) Therefore ought the woman to have a power over her head because of the angels.”

In the public worship of God, a woman wears a veil to show her subjection to the husband as part of God’s divine order; this hierarchy is even acknowledged by the order of angels.
 
But there may be another reason to for wearing the veil.  St. Paul continues,
“(11) But yet neither is the man without the woman, nor the woman without the man, in the Lord.  (12) For as the woman is of the man, so also is the man by the woman, but all things of God.” 

In the beginning, Eve came from Adam and so there is a hierarchy of man over woman; but as now every man comes from the woman so the woman wears a veil to indicate a mystery.

For these reasons, St. Paul tells us that God has given a natural veil to the woman.
“(13) You yourselves judge : doth it become a woman, to pray unto God uncovered? (14) Doth not even nature itself teach you that a man indeed, if he nourish his hair, it is a shame to him?  (15) But if a woman nourish her hair, it is a glory to her; for her hair is given to her for a covering.” 

Finally, should anyone find reason to argue differently, St. Paul writes, 
“(16) But if any man seem to be contentious, we have no such custom, nor the Church of God.”

Following St. Paul and the ancient practice of the Church, the 1917 Code of Canon Law required women to wear veils at liturgical functions,
“Men, in a church or outside a church, while they are assisting at sacred rites, shall be bare-headed, unless approved mores of the people or peculiar circumstances of things determine otherwise; women, however, shall have a covered head and be modestly dressed, especially when they approach the table of the Lord.” (Canon 1262, para. 2)

In recent times the head covering is not a common practice.  What happened?  In 1976, the Sacred Congregation for the Doctrine of the Faith issued the decree, “Declaration on the Admission of Women to the Ministerial Priesthood” (Inter insigniores) which outlined why women cannot be priests.  In this decree we read,
 “... it must be noted that these ordinances, probably inspired by the customs of the period, concern scarcely more than disciplinary practices of minor importance, such as the obligation imposed on women to wear a veil on the head; such requirements no longer have a normative value.”  

Since that time, the new Code of Canon Law (1983) omits the requirement for head coverings... thus, today, in the Church at large, wearing of the head veil by women at holy Mass is rare. 

This ought not be so.  And I think if the matter were better understood, women would find in wearing the veil a dignity good for them individually and good for society.

In explaining my opinion, I will not address wearing the veil for reasons of modesty or subjection, but only as it relates to a great mystery.

St. Paul tells us that woman came from man and as man came first so he has a headship over the woman... thus she wears a veil to reveal her submission to him.  It is also true, however, that since Adam every man now comes from a woman and she should wear a veil for this reason too.

In every conception, the divine Visitor enters the woman alone to create a new person  Only in her does God descend and, like another incarnation, touch the womb, and implant a new and immortal soul.  Now this is a great mystery.  And how do we show mysteries?  We veil them.

A mist covered the earth at its creation; smoke veiled Mount Sinai when Moses received the tablets; a cloud received our Savior at His Ascension.  The sacred is veiled so that we might be drawn to a deeper reality.  At holy Mass, the Tabernacle is veiled because it contains God and the chalice is veiled for it will contain God.  Just as there is a great mystery of God becoming present on our altars - which we “see” by Faith - so covering the woman by a veil makes her life giving role and unique dignity more evident... only she has been chosen as a vessel for new life.

But what of those women who cannot bear children, the old or young or sterile, should they wear a veil?  They should.  For the old or young or sterile still share the nature of woman which is a nature identified with bearing new life... and natures do not change.

If we follow this logic of wearing the veil which points to the mystery of the woman, then we can properly understand the churching of woman after childbirth.  This sacramental is not only an act of thanksgiving but of purification.  Now the purifying is not to cleanse a dirty object, but to cleanse a holy one before it is returned to a common use.  At holy Mass, after Communion, the priest purifies the chalice.  He does this not because the chalice is dirty, but because God has been there.  So the woman is purified, not because she is dirty, but because God had entered her, touched her womb, and through her put another immortal soul into this world.

To close, in my opnion, all women should wear a head veil at holy Mass as a visible witness of her unique privilege and dignity.  If all women understood themselves better, I believe they would value themselves more and appreciate more their unique maternal nature... too much abused, abandoned, and warfared against in our day.  Moreover, if woman recognized her unique privilege, she would defend her dignity by protecting herself against immodest dress, by avoiding bad companions and places, by hating to be reduced to a toy for man’s lower pleasures.

Of all reasons, if all women understood the sacred mystery of their womanhood, they would wear the veil at every holy Mass happily, eagerly, and with honor.  And great good would return to this poor world too much in the dark about the true purpose, happiness, and goal of life.