I think by supplement I really mean replace. I spent a lot of time agonizing over the first chapter of Kittel as he introduces a bunch of concepts such as Bravais lattice and he doesn't clearly define them. It's frustrating and infuriating considering that I have no background in crystallography nor solid state physics, and as an introduction to the subject I haven't found the book to be very helpful.
The only other books I can personally recommend would be Ashcroft and Mermin, Solid State Physics, or Chaikin and Lubensky, Principles of Condensed Matter Physics. Both books are more advanced than Kittel.
Good luck and don't give up! Sometimes a chemistry book may help. -D