Camera follow player when player rotates in y-axis

Hi. I am using the following code in my camera to follow a player object. The camera rotates to the direction the player is moving. The rotation is smooth for all axes except for when the player rotates in the y-axis. I want to have the camera follow animation smooth when the player moves on the y axis as well.

void LateUpdate () {
		//transform.position = Vector3.SmoothDamp(transform.position, (target.transform.position + offset), ref velocity, smoothTime);
		Vector3 destination = target.transform.rotation * offset;
		destination += target.transform.position;
		//transform.position = destination;
		transform.position = Vector3.SmoothDamp(transform.position, destination, ref velocity, smoothTime);

		//transform.LookAt(target.transform.position, target.transform.up);
		LookAtTarget();
	}
	

	void LookAtTarget(){
		Vector3 lookPos = target.transform.position - transform.position;
		Quaternion qt = Quaternion.LookRotation(lookPos, target.transform.up);
		transform.rotation = Quaternion.Lerp(transform.rotation, qt, rotatingSpeed * Time.deltaTime);
		//transform.rotation = qt;
	}

The below gif will give you a better idea:
150609-final.gif

Hi @Veeresh01! I tried your code and for me, the camera rotation is smooth for all the axes. I wonder if the problem is that it is rotating too fast (i.e maybe the rotation in the y axis is bigger, so that is way it does not seem as smooth as the other axes)? Unfortunately, in the gif I cannot appreciate very well the behavior you want to achieve.