"My soul rests in God alone" (62:1), and then, "My soul, wait in silence for God alone" (62:5). If his soul already rests, why tell it to wait?

The psalm doesn't explain. It says both, in that order, and between them come the men who assault "a man": "How long will you assault a man? Would all of you throw him down, like a leaning wall, like a tottering fence?" (62:3), and "They fully intend to throw him down from his lofty place" (62:4). In 62:1 rests is the noun dumiyyah, which Brown-Driver-Briggs gives as "silence, still waiting, repose" (tap rests); in 62:5 wait in silence is a command, domi, from a verb Brown-Driver-Briggs gives as "be or grow dumb, silent, still" (tap silence). Other psalms speak to the psalmist's own soul too: "Why are you in despair, my soul? Why are you disturbed within me? Hope in God!" (Psalm 42:5). Psalm 62 doesn't say what happened between 62:1 and 62:5, and these notes don't say it for it.