"Can I use another transition matrix?"

No, there is a unique transition matrix from one basis to another.

Think about a basis just being some vectors that act like coordinate axes.

If you have another basis, it is like placing another different set of axes on top of the first axes.

Any point in space can then be thought about relative to one set of axes or the other. The relationship between those axes is unique.

If the relationship between the axes is unique it means that the relationship between the basis vectors is unique. Since the transition matrix represents the relationship between the bases then it must be unique.