Slick and unruffled
Strikes me that you can appear composed, cooler than cool, in one of two ways: Attentive, or just acting. Forget the acting bit. It's empty and tiresome. Just ask any actor.
The attentive mind is the best chance we have to achieve composure. Being attentive gives us a chance to intervene instead of letting the primal mind rule over us uncontrolled.
Here's what happens when
we're not attentive.
Information arrives in the brain. It's all 'news' of some kind: whether from the senses, or our thoughts, or memories.
The primal mind receives the information quickest, and processes it quickest. It reacts to it by relying on impressions, estimations or pre-judgments. These are likely to be inaccurate because they are rapid, filling in the gaps where facts are not clear.
An emotion or instinct is instantly activated. It is a response, a signal to motivate us strongly towards reaction or attitude.
Reactions are powerful and rapid, geared towards the short term only. Actions are impulsive, happening almost 'without conscious thinking'.
Actions and reactions are taken impulsively and rapidly, without due consideration of actual facts, fairness, context, or alternative options.
The logical mind has only begin to receive and process information and think things through calmly, looking for facts, truths, and considering things in context. But it's too late by the time it has finished its work.
The Mind's Eye: a tool of awareness.
We can intervene in how we react to things, if we use the main tool we have at our disposal: the ability to look at what's going on in our mind, consciously. When we are attentive, we use a sort of 'inner eye' to observe what is going on. Let's call this 'The Mind's Eye. The attentive mind is in control of the Mind's Eye: being attentive gives us the ability choose what to be aware of. In this way, the attentive mind uses the mind's eye as a tool. Using the mind's eye means we can observe what the mind is up to.
Observing then gives us the to then intervene. Intervention is when we insert wisdom into our actions and reactions. It is what makes us more composed.

Wisdom and composure are the result of combining and choosing from of all aspects of our thinking: primal, logical, and attentive.
In other words, by being attentive, you are building the path to becoming more aware of yourself, and this is essential in the path to becoming more composed. We all know about wisdom, but we struggle to keep it foremost in our minds, to act on it when we really need to, before our impulses and poor judgments have taken hold.
Most of us are quite wise within ourselves but unable to hold on to it from moment to moment: we recognise the wisdom when we hear it, and we understand the sense behind acting calmly, with logic, fairness, humanity and greater things in mind. We know what to do, but don't do what we know. We can act more logically, more fairly, or we can still go with our primal reaction if we think it is appropriate. It takes practice, and it definitely gets better the more we do it.