Remember from Type Dynamics that each of one of us either extraverts our Judging (T or F) function or our Perceiving (S or N) function—and we introvert the other of these functions.
The last letter of your Type (J or P) indicates which function you extravert. This is therefore the function the world hears and sees from you most often (even for introverts). Example: I am an INFP, so I naturally extravert or show the world my Perceiving function (in my case, iNtuition). When I engage with the world around me, I most often show my iNtuition (ask questions, show curiosity, see and talk about connections and ideas, brainstorm, et cetera).
And that is the key to answering this “coaxing” question, because if someone coached me just to “be more extraverted”, that would be unhelpful, awkward and even stressful for me. On the other hand, if they coached me to “share some of your ideas about the future” or “tell us if you see any patterns or trends here” (both iNtuitive), I would much more easily show and share more of my iNtuition (which would look and sound more Extraverted).
As the MBTI practitioner / coach, you can keep all this Type Dynamics complexity “behind the curtain” with you. Just remember that the smoothest way you can get Introverts to show more Extraversion is to ask them to flex their extraverted function. They already share that function in the outer world, so asking them to bring out more of what already comes naturally is a great way to coax Extraversion from an Introvert.