Being centered throughout every phase of the swing can be improved with practice. You cannot teach balance, you can only train it. Your body maintains its balance by reflexes. If you stumble, for example, your body instinctively tries to right itself very quickly. Stand on one leg and bring the other up like you are marching.
Most people rely very heavily on their eyes to maintain their balance.
Ideally you should be able to maintain your balance very well with your eyes closed. Basically, anything that challenges your balance can help improve it. There is no such thing as balance that is too good. You can use the above tests in your balance training. Work up to 2 minutes on each leg.