^^ I have had an outline for such a book sitting on a shelf for years; maybe I'll blow some dust off it one of these days... anyway:
Strangely, there might be less than you would think.
Self-awareness is a state of mind that is not so much triggered by exercises as it is welcomed and nurtured by a mind interested in (or suspicious of) its presence in reality. Unlike critical state tests (RC's) which ask simple yes/no questions (i.e., Am I dreaming?), techniques for assembling self-awareness must by their nature be much more nebulous, more ongoing, and more centered on teaching yourself to float thoughts like "I am," thoughts that are simply not conducive to stimulus/response techniques. So the effective self-awareness techniques you do come across tend to be years-long -- sometimes life-long -- programs or disciplines like
Vipassana yoga, and not short, ready-made actions like RC's. That said:
You might not be able to increase self-awareness while doing decidedly rote things like driving or teeth-brushing, but you can tap it during such mental down times. For instance, think about what you're doing,
literally, when you are driving: notice that you are hurtling down a road at speeds unimaginable just a short century ago, that you are altering your place in your local reality every second, and that you are piloting a couple tons of focused kinetic energy that could change reality for many, including yourself, if your piloting is poor. Recognizing your presence in the moment of driving your car might take an extra few seconds, but that presence can prove profound, once discovered.
Simple chores like brushing your teeth might offer fewer opportunities for to spot your presence in the moment, but even those offer a hint: When you are brushing your teeth, you are generally standing at a bathroom mirror looking at your reflection. If you can use that mirror to take a moment to observe yourself going through the process of simple hygiene, then you might be able to assemble a moment of real personal discovery, a moment that has you wondering things like how you got to that mirror,
why you are brushing your teeth every day, who that person is in the mirror, and any number of other things.
Self-awareness is all about acknowledging, experiencing, and accepting the presence of "You" in any given moment, and associating that presence with the local reality with which it is interacting. Unlike RC's, which demand constants (i.e., your finger cannot pass through your palm in waking-life), the nature of that presence, as relevant to its interaction with reality, is in constant flux. Self-awareness exercises are infinitely variable, and always dependent on the current interaction between You and your local reality... so the technique to trigger self-awareness, if there were one, would have to change every time you did it (this BTW is why the RRC asks essentially open questions and will tend to be different every time it is done).
So in a sense techniques that help you conjure, massage, and develop self-awareness would be in constant flux, and always be dependent (in a circular sort of way) on your interest in and ability to see and define whatever particular moment you might be navigating. Self-awareness techniques, then, would always be changing and rarely work well from one person to the next without modification. This is why it makes more sense to develop, over time, a mindset that welcomes self-awareness as a part of its being so that self-awareness will become a permanent part of you, rather than something to trigger when needed.
In my opinion self-awareness pretty much equals mindfulness, so yes, mindfulness will increase self-awareness. And vise-verse.