Tertullian was an important figure in the Early Church. He lived in Carthage (Tunisia), North Africa (b. 145-160A.D. - d. after 220A.D.)
I've read Tertullian's teaching on head covering in three places (there may be more).
Tertullian has some very interesting arguments about head covering...and some strange ones. There is far too much to cover (hah!) in this post, so I will just point out a few things. If you want to read the documents, click on the names of his works I listed above.
While Tertullian does go through the reasons to cover in 1 Corinthians 11:5-16, like Clement, he emphasises modesty. His take on the angles in verse 10 (where this whole idea of modesty seems to stem for everyone!) is that fallen angels will lust after women who do not cover their heads as the angels lusted after women in Genesis 6:2.
For if (it is) on account of the angels - those, to wit, whom we read of as having fallen from God and heaven on account of concupiscence after females - who can presume that it was bodies already defiled, and relics of human lust, which such angels yearned after, so as not rather to have been inflamed for virgins, whose bloom pleads an excuse for human lust likewise? For thus does Scripture withal suggest: And it came to pass, it says, when men had begun to grow more numerous upon the earth, there were withal daughters born them; but the sons of God, having descried the daughters of men, that they were fair, took to themselves wives of all whom they elected.
In chapter 22 of On Prayer, Tertullian also links veils to the marriage veil, saying virgins are wedded to Christ.
You do well in falsely assuming the married character, if you veil your head; nay, you do not seem to assume it falsely, for you are wedded to Christ: to Him you have surrendered your body; act as becomes your Husband's discipline. If He bids the brides of others to be veiled, His own, of course, much more.
Tertullian speaks of covering in the church most of the time, but he ends On the Veiling of Virgins with a call for women to veil when Psalms are read or God is mentioned. Here is the second to last sentence of that book.
It is incumbent, then, at all times in and in every place, to walk mindful of the law, prepared and equipped in readiness to meet every mention of God; who if He be in the heart, will be recognised as well in the head of females.
At the same time, Tertullian criticises virgins, in chapter 13 of the same book, who cover in public, but do not cover in church.
...and as they veil their head in presence of heathens, let them at all events in the church conceal their virginity, which they do veil outside the church. They fear strangers: let them stand in awe of the brethren too; or else let them have the consistent hardihood to appear as virgins in the streets as well, as they have the hardihood to do in the churches.
We can get two things from this: 1) that many Christians of the day understood covering to be a custom for church only and 2) that veiling in public was an accepted cultural phenomenon - at least by some if not at all.
Women's Silence in Church.
Tertullian embraced the Montanist movement of his day, which was surprisingly similar to the Pentecostal and Charismatic movements of today. He therefore expected, unlike many of his contemporaries, God's Spirit to move people to prophesy and seemingly saw this as an exception to 1 Corinthians 14:24-25.
In Against Marcion 5:8, Tertullian says this:
In precisely the same manner, when enjoining on women silence in the church, that they speak not for the mere sake of learning 1 Corinthians 14:34-35 (although that even they have the right of prophesying, he has already shown when he covers the woman that prophesies with a veil), he goes to the law for hs sanction that woman should be under obedience.
How much should you cover?
Tertullian believed that a woman should cover down to the top of her robe (i.e. covering the neck). He appeals to an experience a woman had while praying:
To us the Lord has, even by reveloations, measured the space for the veil to extend over. For a certain sister of ours was thus addressed by an angel, beating her neck, as if in applause: "Elegant neck, and deservedly bare! It is well for you to unveil yourself from the head right down to the loins, lest withal this freedom of your neck profit you not!" And, of course, what you have said to one you have said to all.
What do we do with all this?
Once again, the Church Fathers are useful, but are not Scripture. Nothing Tertullian says here is binding on Christians, except in as far as he repeats what Scripture says.
Both Tertullian and Clement emphasise modesty very strongly. They both believe that covering outside of church is a good thing to do, though they do not explicitly draw this from Scripture. They both consider that covering must be more than a small piece of cloth (though they both understand it to be a cloth!)
Both these Fathers also lived in a time when women covering was culturally practiced as well, and this seems to have influenced their understanding of the practice extensively.
They also both appealed to extra-biblical authority.
In other ways, however, they are different. Tertullian did not require women to cover their faces. They interpreted 1 Corinthians 11:10 differently (in terms of the identity of the angels) and Tertullian believed women could prophesy in the church. Tertullian was addressing specific issues, i.e. whether unmarried women should cover or against the heresy of the Marcionites.
With all that in mind, we can take what Tertullian said and use his insights to help us understand 1 Corinthians 11:2-16. We can see how he applied that passage in his own time and what cultural assumptions shaped his applications. Then we let that information prod us to look critically at how we apply 1 Corinthians 11:2-16 to ours.