Get the entire front arm to the level of the shoulders and use physics to swing the bat naturally and efficiently. The torso is the axis around which the bat should swing.
In reality, the swing involves multiple pivot points, which makes things very complex. The barrel actually rotates around the hands, but the location of the hand pivot point is affected by other actions which cause that pivot point to move. The concept of working on a tilted plane around the torso is important, however.