Unless your mesh is extremely coarse right now, refining it will not change the order of your convergence. If you are getting good linear convergence, then it is likely your mesh is fine enough for the method to be asymptotic, and it will never change, no matter how fine your mesh gets (until you bottom out your residuals). I would recommend looking at the analysis of your derivative approximation and see if the assumption is made that the derivative being approximated is continuous.
I may be mistaken about this fact, but if it is based on some polynomial approximation of the function, this may cause the quadratic convergence to fail at singularities. This will drop your convergence from second order to some lower order at the point where you state your coefficient is a step function. Dropping the order at one point, be it an internal singularity or a boundary condition reduces the overall order to that same value.
Also, another possibility I thought of, are you sure that the method is second order global, and not just second order local? It is a simple mistake to make, and would explain the one order lower convergence.
If you give more detail as to what your PDE is and what method you are using to approximate your derivatives, it may help us give a more specific answer as to what could be causing the incorrect convergence rate.