Adding a "Fax Number" when adding a new contact in my iPhone 5, ios6 When adding a new contact, I"m confused when I get to the "phone contact numbers".

I only see Mobile and Home.  I want to add a fax number, but it isn't available and it isn't offered in Add another field.
On your contact, click on the label type : 



And it will give you access to the label types. 

These are iOS 7 screens, but it's the same on older versions !