I am a PhD student in mathematics starting my third year with the goal of a tenure track position at an R1 university. I believe I have enough research completed that I could graduate this year if I chose to, however my funding situation allows me to stay in my program for up to 6 years fully funded and with little teaching obligation (I do have the opportunity to teach a full course over the summer and plan to do so). I am currently planning to stay for all 6 years, unless I receive a very good offer in my 5th year. I have somewhat different taste in problems from my advisor, so I am currently working on becoming a more independent researcher.
My question is on how to best take advantage of the remaining years of my degree. Is there anything more subtle than "do more research"? What can I do to prepare myself for a postdoc or otherwise get ahead? Are there things that can be done as a graduate student in this situation that cannot easily be done as a postdoc or that are much better to start earlier?