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"?
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.