Mouse Look from any Orientation

I am working on a first person game where the character is able to latch on to any flat surface and walk on it (kind of like moon boots). I am having issues with the camera since most implementations of mouse look I have been able to find depend on the player being oriented straight up and down in the world, so as soon as my "flipping" animation (the player orienting to the new surface) has finished, the camera will instantly flip back to straight up and down as defined by the world. What I need is a way to implement mouse look so that it does not reset the rotation after my animation. I am currently using:

transform.Rotate(-Input.getAxis("Mouse Y") turnSpeed Time.deltaTime, Input.getAxis("Mouse X") turnSpeed Time.deltaTime, 0);

in order to perform my mouse look rotations while avoiding the standard method, but this is clearly incorrect since I get some weird rotations when turning the camera horizontally. Is there a better way to implement mouse look so that it works correctly regardless of the orientation of the player?