At last! Now we enter into how salvation works. However, the previous two chapters depicting the nature and all pervasiveness of sin are essential. No one can come to Christ unless he acknowledges his own sin, no one can appreciate the work of Christ unless he acknowledges his own sin. The work of Christ only makes sense in the light (or the darkness) of our own sin.
The five verses in 3:21-26 have been described as the most important paragraph ever written.
The “but now” can point to a logical and a temporal sequence. In time the appearance of Jesus showed how this new righteousness works. Logically it follows on from the Law.
The Jews sought to be righteous by obeying the law, but now a different righteousness has been made known. “the law” in the first phrase does not have a definite article in the Greek and is referring to law in general, though obviously with Law uppermost in Paul’s mind, but the we will not find righteousness on the principle of law. Have you noticed that when something serious goes wrong anywhere, whether in society as a whole or in the organisation you work for, what happens is this. There is an inquiry, the inquiry makes a multitude of recommendations, then new procedures, rules or laws are introduced. And nothing really changes, a few months or years later another disaster will occur. Law cannot bring righteousness.
Now while this righteousness is not by the Law, it is entirely consistent with the Law. It is what the Law and the Prophets were all pointing towards. The Old Testament can be summed up as saying sin is the problem, you are incapable of living Godly lives, but there is a solution coming. In Christ that solution has been revealed.