You might consider using the Nonlinear platform instead of Fit Curve. Both fit nonlinear models, but Nonlinear provides more detailed options for controlling parameter estimation, among other things. (Fit Curve is for simplified out-of-the box nonlinear modeling.) One of the Red Triangle options in Nonlinear is declaring lower or upper bounds on parameter estimates, which you can use to address an asymptote being too high or low.
Note that Nonlinear requires a model equation specified as a column formula in your data table. It can write one for you if you're using one of JMP's built-in models. In the launch dialog, just use the Model Library button to declare a Logistic 4P model to get started.
Ross Metusalem
JMP Academic Ambassador