Something is definitely off about your recollection here. g(x) = 1/√(1+x²) is always positive, whereas tanh(x) is negative for negative values of x. So regardless of your f, g(f(x)) can’t ever be a correct approximation for negative x.
Looks like f(x)=x+x^3/5 gets you two decimal digits everywhere [1], and you could improve that with more terms in the polynomial.
Edit: you can derive f(x) by doing a series expansion of tanh(x)/√(1-tanh(x)²) about x=0, and Mathematica tells me the first few terms are
f(x) = x + x^3/6 + x^5/120 + x^7/5040 + O(x^9)
And yes, the Taylor series is pretty good but you can usually squeeze a bit more accuracy-over-range by, eg, using an optimizer for the coefficients. Also, note that in your notebook you have x^3/5 but x^3/6 is probably what you wanted.