Controller Assignments assume a hardware input of 0-127 and automatically scale this range to the working range of the particular software parameter. You have three parameters to tweak the working range: Min/Max, Multiply and Mode.
An example for the Logic Tape Delay LFO
The LFO has a MIDI range of 1-100. Means, the full hardware fader range of 0-127 will be squeezed into 0-100 to match the LFO range to the full fader range. If you need a minimum value you can raise the starting point by increasing the Min value in the Controller Assignments panel. And to set a desired maximum value you can change the Multiply value to something lower than 1.00
Setting the minimum to 40 and the multiplication to 0.80 will give you a parameter range of 0.52-6.40 Hz for the LFO, instead of the full range:
Hardware fader fully down (value 0):
Hardware fader fully up (value 127):
Changing the Mode to Direct does change the range but using the multiplication alone is normally the better way.
The drawback is that a minimum value greater than 0 will give you a dead lower zone on your fader. You may try to avoid this by leaving minimum to zero and multiply by something higher than 1.00 but this makes your fader actually more sensitive.
You can even control multiple parameters with one fader in a certain manner by setting different ranges for them. For that you may want to increase the allowed parameters for one hardware control in the preferences.
If you don't want the dead fader zone or need to apply a curve for better control, you cannot use Controller Assignments. In this cases you have to work in the environment and manipulate the incoming MIDI according to your needs. This is more complicated though.