Use the arm to move the hand and fingers not just across strings but also from low to high positions and back.
This exercise takes the arm from where it is most out to the left (1st position on E-string) to where it is most turned in to the right (high positions on G-string).
When you shift up high there will be a bend at the wrist, make sure that when you shift down your hand comes back in line with the forearm. Try stroking the side of the violin with the palm of your hand on the way down.
Supporting the violin with the right hand encourages the development of a stable violin hold around which the hand can move freely.
Resources - Bibliography
Young Strings in Action by Paul Rolland, Action 2: The Shuttle Between Registers