You don't need to write your own code to achieve the effect you are looking for.

I'm not sure which way to interpret what you're asking, but I'll my answer for each:

1) You're trying to rotate the object so that it doesn't pivot around the Y axis.

transform.LookAt(target, newAxis); // Change "newAxis" to the axis you want.

2) You want to make it so that it is not necessarily the forward axis of an object that looks at another object. You might want the side to be angled towards the target.

transform.LookAt(target);
transform.Rotate(eulerAngleOffset, Space.Self);

eulerAngleOffset is the rotation in local space applied AFTER the transform is rotated to look at the target. For example, to orient left side of the transform towards the target instead of the forward side, eulerAngleOffset would be:

eulerAngleOffset = new Vector3(90, 0, 0);

Top would be:

eulerAngleOffset = new Vector3(0, 90, 0);

To roll the camera to the side:

eulerAngleOffset = new Vector3(0, 0, 90);

These functions can be broken down into operations with Quaternions. When you can wrap your head around them, they are extremely useful. I'm not sure of what the source code is for the quaternion functions, but here is an example of code that "points" an arbitrary local direction of an object towards an arbitrary global direction, taking the shortest way around possible:

Vector3 fromGlobalDir = transform.rotation * fromLocalDir; // Same as Transform.TransformDirection().
transform.rotation = Quaternion.FromToRotation(fromGlobalDir, toGlobalDir) * transform.rotation;