The need for phase wrapping?
I'm not sure if I understand your problem, but I'll give it a try. If you have a complex frequency response

$$H(\omega)=|H(\omega)|e^{j\phi(\omega)}\tag{1}$$

where $\phi(\omega)$ is the phase, then you probably know that for a given frequency $\omega_0$, the frequency response might as well be written as

$$H(\omega_0)=|H(\omega_0)|e^{j(\phi(\omega_0)+2k\pi)}\tag{2}$$

where $k$ is some integer. The question is, when you compute the complex value $H(\omega_0)$, which of the infinitely many possible phases do you choose? The convention is to choose the principal value which is in the interval $(-\pi,\pi]$. The consequence is that whenever you compute the function $\phi(\omega)$ for continuous $\omega$, you'll get jumps as soon as the phase hits $-\pi$ or $\pi$, but no matter how you define the phase, you will not be able to avoid jumps. So if you want a continuous function you need to do phase unwrapping.