What should I do to make the most of a stopgap postdoc position offered by my PhD supervisor?
Your #1 goal is to find a real post-doc position, so think of why that hasn't happened yet if you have been out trying for a while. 

Next is finishing up those papers, since that may help with the #1 goal. Further, once you do get another position where you want to chase after new things, you will find that time slips away. 

Anything else is gravy. Do not let anything new (much less competitions for goodness sake) get in the way of those. Don't think of it as 6 months, think of it as week-to-week until you have accepted a job offer.