In addition to "the belt which is truth" the apostle Paul tells us to put on "the breastplate which is righteous" (another genitive of apposition). In military terms, this was an important piece of armor that protected a vital part of the body from great harm. A wound here was very serious, possibly fatal.
The "righteousness" that Paul has in view here, is not the "righteousness of Christ" that is imputed to us by faith. It is a moral, ethical righteousness which is the inevitable result of being joined to Christ. This righteousness is what Paul refers to elsewhere in Ephesians as "holiness" (4:24-25) or "goodness" (5:8-9).
Without this kind of righteousness of life we can be seriously wounded in the battle. Holiness, or conformity to Jesus Christ, is essential in our lives as Christians. Where it is lacking we are weak and unable to fight as we should. Or we will be distracted and not wholeheartedly devoted to Jesus. Those who have the "righteousness" of Jesus transferred to their account (i.e. imputed righteousness), must walk or live as Jesus did (i.e. ethical righteousness). The two are bound together in the gospel. We cannot have one without the other.
Great harm has been done to the cause of Christ by "unholy" people. Christians who take "unrighteous" stands on insignificant issues. Or those who make accusations in ignorance. I never cease to be amazed at how some "strain out gnats but swallow camels." Surely the righteousness that Paul is speaking about in connection with spiritual warfare is not mean-spirited, weird, hokey, or just plain silly. Thinking people are often turned off Christians and churches who think they are "standing firm" when in fact they are torturing the Christian gospel and remaking it in their own miserable and paranoid image.
Conversely, great good has been done by those who have submitted their lives to the Scriptures and the gentle but powerful teaching ministry of the Holy Spirit. Such people are a delight to be around. They bring people together instead of driving them apart. They are patient, compassionate and know how to leave room for God to do his work in their hearts and in the hearts of others. This kind of personal and ethical conformity to the likeness of Jesus pushes back the reign of darkness and allows a little bit of heaven to be seen on earth. It is like a ray of sunshine breaking through dark clouds as a testimony to God's existence and grace.