I know covering the heads does not make a person a better Christian. Many of our traditional churches have a vestige of this teaching that has remained in the popular custom of women wearing hats to church on Sunday. God looks at the heart of people.
If that is what your church wants you to do then go with it. If the church does not observe this trandition, then don't worry about it.
What is God's real meaning of these verses and what is the true "covering?" One way we can discern between false and true teaching is to examine the fruit of it. Does it bring freedom, or legalism and bondage?
In conclusion, God is our covering spiritually, for both the man and the woman. Both husbands and wives should cover one another in prayers and with their devotion and loving deeds. The Lord is always teaching us about heart attitudes in the New Testament of the Bible and many of the parables of Jesus and the illustrations of Paul were used to point out these very things. Paul's conclusion to the question of women veiling in I Corinthians 11:16, "But if any man seem to be contentious, we have no such custom, neither the churches of God." He simply says this is a custom of the women, but it is not a church ordinance. Therefore, if in some cultures, the women are still covering their heads and it is their custom, it would not hurt us to honor that custom if we go there to preach by covering our head if we are a woman. (This could also pertain to our dress as western women when we are in a foreign land. We should restrain from wearing short dresses and shorts. Personally, I don't think wearing skimpy clothes is a good witness, even in our own country, as it doesn't represent Godly dress.) The people we are preaching to in foreign lands might be more open to receive our message if we are considerate of their customs. Paul made this statement:
1 Corinthians 9:20-22 "And unto the Jews I became as a Jew, that I might gain the Jews; to them that are under the law, as under the law, that I might gain them that are under the law; 21 To them that are without law, as without law, (being not without law to God, but under the law to Christ,) that I might gain them that are without law. 22 To the weak became I as weak, that I might gain the weak: I am made all things to all men, that I might by all means save some."
If a person is more comfortable in covering their heads because of what your church has taught then do it. If a person is not comfortable because it was not taught to them because of a different type of church then, that is ok too! It is what makes you comfortable but it has nothing to do with salvation. Salvation is a heart issue. It is not sin to not cover your head because Jesus is our covering. These are traditions of man.