I need a little help with headbobber script (C#)

Hi! I used the headbobber script from @thatCamelCode. The script is awesome, but there is one little problem. Default camera position on Y is 0.6f, and when the player starts to walk, camera immediately change position on Y to 0.69f, not smooth. Tiny problem, but still want to headbob began smoothly. Any ideas? P.S. Sorry for my English :) @thatCamelCode headbobber code:

 public class Headbob : MonoBehaviour {
 public Vector3 restPosition; //local position where your camera would rest when it's not bobbing.
          public float transitionSpeed = 20f; //smooths out the transition from moving to not moving.
          public float bobSpeed = 4.8f; //how quickly the player's head bobs.
          public float bobAmount = 0.05f; //how dramatic the bob is. Increasing this in conjunction with bobSpeed gives a nice effect for sprinting.
          
          float timer = Mathf.PI / 2; //initialized as this value because this is where sin = 1. So, this will make the camera always start at the crest of the sin wave, simulating someone picking up their foot and starting to walk--you experience a bob upwards when you start walking as your foot pushes off the ground, the left and right bobs come as you walk.
          Vector3 camPos;
      
          void Awake()
          {
              camPos = transform.localPosition;
          }
      
          void Update()
          {
              if (Input.GetAxisRaw("Horizontal") != 0 || Input.GetAxisRaw("Vertical") != 0) //moving
              {
                  timer += bobSpeed * Time.deltaTime;
      
                  //use the timer value to set the position
                  Vector3 newPosition = new Vector3(Mathf.Cos(timer) * bobAmount, restPosition.y + Mathf.Abs((Mathf.Sin(timer) * bobAmount)), restPosition.z); //abs val of y for a parabolic path
                  camPos = newPosition;
              }
              else
              {
                  timer = Mathf.PI / 2; //reinitialize
      
                  Vector3 newPosition = new Vector3(Mathf.Lerp(camPos.x, restPosition.x, transitionSpeed * Time.deltaTime), Mathf.Lerp(camPos.y, restPosition.y, transitionSpeed * Time.deltaTime), Mathf.Lerp(camPos.z, restPosition.z, transitionSpeed * Time.deltaTime)); //transition smoothly from walking to stopping.
                  camPos = newPosition;
              }
      
              if (timer > Mathf.PI * 2) //completed a full cycle on the unit circle. Reset to 0 to avoid bloated values.
                  timer = 0;
          }
      }

I know that it is necessary to change this code,

 Vector3 newPosition = new Vector3(Mathf.Cos(timer) * bobAmount, restPosition.y + Mathf.Abs((Mathf.Sin(timer) * bobAmount)), restPosition.z);

but do not know how to have a camera at the beginning of the shift position smoothly.