Does the Bible teach a doctrine of "covering" (that one must be "covered" by a spiritual authority higher up in the spiritual chain of command in order to be protected, blessed, etc.)? I see no such doctrine in Scripture.
The Bible does not teach a specific "covering" doctrine, at least not in the way you described it. Rather, the Bible teaches that all men and women are under authority, both in the church and in society, and therefore Christians are expected to obey and respect all authority.
Specifically, the body of Christ should honor the leaders in their life. For example, Hebrews says:
Heb. 13:17 Obey your leaders and submit to them, for they keep watch over your souls as those who will give an account. Let them do this with joy and not with grief, for this would be unprofitable for you.
Likewise, Paul admonishes the church to submit to human institutions of authority in Romans 13:
Rom. 13:1 Every person is to be in subjection to the governing authorities. For there is no authority except from God, and those which exist are established by God.
Rom. 13:2 Therefore whoever resists authority has opposed the ordinance of God; and they who have opposed will receive condemnation upon themselves.
In both cases, the writers connect our obedience with spiritual profit. The Hebrews writer says if we make it difficult for our leaders to fulfill their responsibilities, we will be the losers. Our leaders' skills in teaching and counseling us will ultimately lead to a better outcome on our judgment day in eternity. On the other hand, if we resist authority in the church, we may fall into a state of rebellion and sin, leading to an unprofitable outcome on the day of our judgment (see 1Cor 3).
Similarly, Paul says that if we make a habit of resisting human government, we are likely to feel the sting of human justice, since those who oppose human government are opposing God Himself.
In general, godliness in life will bring a blessing while ungodliness results in consequences, including potentially divine judgment. Nevertheless, we are concerned anytime men promise "blessing" to those who adhere to their rules. That kind of thinking eventually leads to cults. The Bible's command that we submit to human authority is not intended to set the rule of men above the law of God.
Ultimately, Christians must submit to the authority of the Spirit and God's Word. These trump all earthly authority in matters of righteousness and godliness. When the expectations of the Spirit and the Word conflict with the demands of men, we must obey God and endure the consequences that follow (just like Jesus did).
