Our conscious experience is that we're conscious every time we're conscious of it. This is probably because we're not conscious of the times we're not conscious. :P
That said I'm not sure that being actively conscious should really be a goal. Does it make us happier? More effective? For a lot of things I find myself trying to avoid becoming self-conscious because it interferes with my ability to do the thing I'm trying to do.
Yes, this kind of experience exists. However, in my experience, the "self-consciousness" is actually not the same "consciousness" I am talking about, but an experience related to a stage in learning. When you become a little bit aware that you cannot do something, it might temporarily make you anxious. Then, to overcome this anxiousness, you try to use your analytical mind to control your body (in case we are talking about some physical action), and it does not work really well, so you become even more anxious, creating a vicious loop.
Analytical mind is actually not very good at micro-managing the body. You can test this by trying to do something very simple, such as lifting a spoon, and doing it with your analytical mind and without interference from your analytical mind. If you really try this experiment (thinking about it is not enough, you really need to do it yourself), you can learn about the limitations of your analytical mind by yourself.
Your experience can also be expressed differently. What you have experienced as consciousness is actually your analytical mind, and by "avoiding becoming self-conscious" you are trying to let your body (or some other part of you, depending on what you try to do) to do it's thing without micro-management from the analytical mind.
The book "Inner game of tennis" explains how olympic-level athletes (in this case tennis players) need to learn to let go of this control by the analytical mind in their practice. When you can let go, and you are aware of it, the end result is an experience of flow.
For me, being in the flow state feels good, and I can do things that would not be possible otherwise (in my case in dancing). It is something I aim to experience more in my life.
Does this make us efficient? You cannot talk about efficiency separate from your aim. For example, it may be time-efficient to eat all your meals at McDonald's, but the long-term consequences of that choice could make you less than efficient on some other aim you may have.
Happiness is a difficult concept as well. I have experienced more happiness through increase of flow in my life. I would say that experiencing more flow is one of the ways to happiness. But depending on your definition of happiness, you may disagree with this.
That said I'm not sure that being actively conscious should really be a goal. Does it make us happier? More effective? For a lot of things I find myself trying to avoid becoming self-conscious because it interferes with my ability to do the thing I'm trying to do.