After some thought, I think it's fair for me to offer this:
One duty of a clan head is to train his heir. It's unstated at this point whether Hazō or Noburi is the heir (Keiko definitely is not interested), but it would be perfectly reasonable to think it was Hazō. He could therefore go to Jiraiya and ask to be brought into some high-level meetings with orders to sit silently against the wall and observe, or help with paperwork, or etc.
I'm not saying he would go for it at all, and depending on what you ask for he might tell you that you're not ready. Still, there are undoubtedly non-classified things that he does as Hokage that he would be willing to let you in on.