In libgmp I think the numbers have to be around 100,000 bits or 30,000 digits for FFT to be faster.
https://gmplib.org/manual/Multiplication-Algorithms
GMP has lots of other methods in between schoolbook multiplication and FFT multiplication. A nice one is Karatsuba multiplication which is very easy to understand and delivers O(n^1.58) rather than O(n^2) performance. Python uses this method for multiplying large numbers together