People often believe that they are unable to start the day until they have had their first coffee, but the truth of the matter is that this is the wrong thing for us to do.
Instead, when we wake, our brain is already very active at helping us to get our energy levels up to a point where we feel that we can actually function. It does this by releasing cortisol that then floods our system. This cortisol is a natural stimulant that does indeed help to fire us up for the difficult day ahead, so our body does not need the caffeine in the way that we believe.
However, if you still feel that this should be part of your morning ritual, then the best way in which you can feel the absolute benefits of it is to wait until later in the morning before you go ahead and have that cup. You will feel that it helps you more than if you had consumed your first coffee as early as possible.