Apologies if my post came across in the wrong way. All I was trying to say is that, when our playing is efficient and properly executed, that we shouldn't have to worry about range - the range comes naturally as we progress (I'm sure your teacher will be saying the same if you ask them). Range shouldn't be something that we 'work up to' or set as a goal as mentioned in your original post. (IMHO)
I've always seen range as a 'side-effect' of correct and efficient playing. Relax, and enjoy the learning process with your teacher and the range will come. Your teacher will keep you playing pieces that will gently stretch your abilities and as time goes on, the C will come. It will take time though - it may take 6 months, it may be 2 years, but the notes will come when they are ready.
Hope this helps in some way.
(P.S. the only reason I refer to the note as just a C is that I've found in my teaching and in my own learning, that calling them low C, middle C high C, double-high C, triple high C etc, can sometimes lead to a psychological block - if we think of the notes as all the same, there's no 'stigma' attached to them. A double high C becomes just another C like the rest of them, but again eveyone is free to call the notes what they want, just as long as it makes sense to them.)