Solved
Not exactly sure what you mean by "The combination of these three functions gives a family of non-asymptotic sigmoid functions". Which kind of "combination"?
If you have a simple additive combination in mind, you would have to do it that way, nut the result sure is not satisfactory (and very sensitive wrt the guess values):

But if you want to say that you would like to split your curve into three parts and fit a different function type to each part, then you would have to do the splitting yourself and fit three times.
BTW, Here's a function which seems to fit your data pretty well:

Enter your E-mail address. We'll send you an e-mail with instructions to reset your password.

