How does Bigger overcome fear in Native Son?

The tragedy of Bigger's story is that he has no outlet for overcoming his fear other than aggression and violence. In some ways, it's surprising that the first section of the book is titled "Fear" because this implies that this is the emotion governing Bigger's personality and actions. Bigger isn't afraid in a physical sense. His killing of the rat in the opening scene demonstrates this, though it's a trivial thing in itself. Bigger's fears are not physical, but psychological. At the bottom is his insecurity, a feeling of worthlessness that has been imposed upon him by the outside world for both racial and class reasons. Even among his friends, he feels a relentless need to prove himself, threatening them with his knife and slashing the baize covering of the billiard table. This is his means of distracting others from his own palpable sense of weakness and insecurity.
It would be simplistic to conclude that he overcomes fear by acting out violence. In the crucial episode in which he accidentally kills Mary, it is his fear of being discovered in the room with her that prompts him to silence her with the pillow. But Bigger's own insight later into his actions is that "he knew that in some sense the girl's death had not been accidental." Bigger realizes that his anger—both directed against others and against himself—has been expressed many times before. A sense of liberation comes to him as a result of the killing. Is it because he always has known that he's living on the outside, a cast-off of the system, and now there is a feeling of release coming from his crime? He no longer has to pretend to conform to the world's demands which have caused his state of fear to begin with.
Of course, the killing may have dispelled one kind of fear, but it created another. In spite of the ransom plan, Bigger probably realizes from the start that the police will catch up to him. The "Flight" section of the novel shows Bigger in a greater state of hopeless fear than in the relatively subdued actions that have occurred before his coming to the Dalton house results in catastrophe.