How do you improve your odds of obtaining a new position without giving away that you are in the market on social media. Since many people in your current work life have access to you social media pages it seems detrimental to the current work status.
Many people use social media to keep abreast of trends and continue to grow / work their network as part of everyday job activities. Being active in the social media sphere does not immediately mean that someone is searching for new opportunities. As you engage with people on Linkedin or on other social sites, you can take the opportunity to discuss your goals, while also learning about the other person’s interests / goals. This will do something for you both: it allows you to potentially uncover an opportunity, grow your network and facilitate friendships.
As with any job search there is going to be a certain level of risk. You cannot avoid this. My point, however, is that being a part of social media extends beyond job search to show that you are an active member of a larger community. This desire / willingness to remain connected is mutually exclusive to that of job search.
Please let me know if that answers your question. I am happy to opine on any other questions you have.