What chocobo race challenge gives Cure III?

[Jeykama]

So I still have a while. I changed my mind and did as many challenges as I could with my current chocobo, getting up to challenge 4 or 5 done. I don't see any new racing manuals though. Do those come from later races, or am I just blind?

When you complete a challenge that awards new manuals, you'll actually get a system message during the victory sequence. At #4-5 you should have gotten some new books, maybe like Paradigm Shift and Super Sprint. They unlock somewhat gradually.