If I understand well, you just need to make a copy of the material for parts of your object so that each part can have its own material and can be changed without affecting the others, right?
Here I have a sphere with a green material applied to it :


To add a new material slot, you just click the +
button on the right :

The new material slot is empty, you can either add a completely new material with + New
or you can chose from the already existing one with the dropdown list (this is what you want if I understood you well) :

You have now two material slots but right now they are currently using the same material and if you try to change one of them, the other one will change :

It's because the two material slots are using the same material data-block and you can see here how many other slots are sharing that same material data-block (Blender refers to them as users
and the little number that you see near the name is the number of users
for the material) :

Here we have 3 users
. To make the current user
have its own copy of the data-block, you can just click on that number :

After doing that, you now have two different materials and you can now edit either of the two without affecting the other one :

Here, I renamed the two materials and changed the color of the second one. (Notice also that the red
material doesn't have the little number near its name anymore since it has now one user
)
Now you can just go to edit mode and select any part of the mesh that you want :

And then, select the material you want to apply to it and click Assign
:


You can repeat this steps for as many parts of the object as you want.