You are fitting a normal distribution, which implies infinite extent.
In any case where the standard deviation is greater than 1/sqrt(2*pi) then no element of the pdf will be greater than 1 and no rescale is required. If the standard deviation is smaller than that then the pdf at the location of the mean will be greater than 1 and any rescale will prevent the integral from working. You would need to increase the standard deviation to prevent that.
However, I wonder whether your data is truly sampling an infinite distribution. I have not looked at the data, but when a set of data is involved it is far more common that the situation involves a finite span... and finite spans are incompatible with normal distribution. I would ask whether perhaps you should be looking for a beta distribution instead of a normal distribution.