You may want to have a look at this:

https://arxiv.org/abs/1706.10219

It is time consuming but high-precision method.

Warning: self-promotion

I presume the "numerical" aspect of numerical differentiation is very well studied, what is little bit forgotten is "statistical" aspect. To increase the precision of numerical differentiation do the following:

1. Chose your favorite high-precision "standard" method based on some step size H.

2. Compute the value of the derivative with the method chosen in 1) many times with different but reasonable step sizes h. Each time you may pick h as a random number from the interval (0.5H/10, 1.5H/10) where H is an appropriate step size for the method you use.

3. Average the results.

Your result may gain several orders of magnitude in the absolute error wrt. the non-averaged result. It is time-consuming method, but we discuss here precision and not time consumption. I franky believe it is the most precise method of numerical differentation at the market today (yet systematically refused by journals)

https://arxiv.org/abs/1706.10219