Does もう modify 泳いだ here, as in "already moved (lit. swam through air/space)"? So that "while X (tearing hair with one hand due to headache), the trembling other hand already moved"?
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The もう here is the same as the one in もう一つ "another one".
In your example sentence, [he] is scratching off his head with one arm. The other, shaking arm [was trying to swim in the air or whatever].
So, もう片方の手 means "the other arm" (as opposed to "the one arm").
This もう can be used with arms, items, people, anything. Like
I saw two men. One wearing a hat, the other one with a walking stick.