I experimented from the starting point I used simply because I felt like trying it, learning as I went. (I'm the author.)

A minimax result is a small improvement on Chebyshev. Though my program is too slow to be good for experimenting with higher-order approximations, for which a Remez algorithm implementation would be far more generally useful.

https://en.wikipedia.org/wiki/Remez_algorithm

I think Remez + the end-point fitting trick I stumbled on would be the practical way to extend the last two parts in the article to higher order approximations. And make it exact instead of only "almost".