You can easily move the Little Green Button around your screen without activating it, by left clicking once on the button and dragging it to a new location on your screen:
Place your mouse cursor over the Little Green Button.
Left click once and hold down the mouse button.
While holding the mouse button, position the Little Green Button anywhere on your screen(s).
Let go of the mouse button when you are happy with the position.
Note: to activate the button you need to double left click on it.