Moving an object in a curve along the z-axis

I'm currently working on following scene:

The idea is for the player to use the "A" and "D" keys to move by the X-axis. Then by using "W" and "S" jump up and down the stairs. I've roughly achieved this with my following code(PlayerController.cs):

 public class PlayerController : MonoBehaviour {
 
     public float playerSpeedX;
     public float playerSpeedZ;
     public float jumpSpeedUp;
     public float jumpSpeedDown;
     Rigidbody rigid;
 
     public bool grounded;
     public bool positionTier1;
     public bool positionTier2;
     public bool positionTier3;
 
     // Use this for initialization
     void Start () {
         rigid = GetComponent<Rigidbody> ();
         positionTier1 = true;
     }
 
     //---------------------------KEY INPUT---------------------------
 
     // Update is called once per frame
     void Update () {
         //If listed keys are pressed, move left.x
         if (Input.GetKey (KeyCode.A) || Input.GetKey (KeyCode.LeftArrow)) {
             transform.position += Vector3.left * playerSpeedX * Time.deltaTime;
         }
 
         //If listed keys are pressed, move right.x
         if (Input.GetKey (KeyCode.D) || Input.GetKey (KeyCode.RightArrow)) {
             transform.position += Vector3.right * playerSpeedX * Time.deltaTime;
         }
 
         if(Input.GetKeyDown (KeyCode.W) || Input.GetKeyDown (KeyCode.UpArrow)) {
             jumpUp ();
             playerSpeedX = 0;
             StartCoroutine (moveLevelTier1 ());
             StartCoroutine (moveLevelTier2 ());
             StartCoroutine(waitWhileJumping());
         }
 
         if (Input.GetKeyDown (KeyCode.S) || Input.GetKeyDown (KeyCode.DownArrow)) {
             jumpDown ();
             playerSpeedX = 0;
             StartCoroutine (moveLevelTier3 ());
             StartCoroutine (moveLevelTier4 ());
             StartCoroutine (waitWhileJumping ());
         }
     }
 
 
     //---------------------------CHECKING IF GROUNDED---------------------------
 
     //If Player is on the ground, grounded = true
     void OnCollisionStay (Collision collisionInfo) {
         grounded = true;
     }
 
     //If Player is in the air (exitin the ground), grounded = false
     void OnCollisionExit (Collision collisionInfo) {
         grounded = false;
     }
 
     //---------------------------JUMP FUNCTIONS---------------------------
 
     //Jump function
     void jumpUp() {
         if (grounded == true) {
             rigid.velocity = Vector3.up * jumpSpeedUp;
         }
     }
 
     void jumpDown() {
         if (grounded == true) {
             rigid.velocity = Vector3.up * jumpSpeedDown;
         }
     }
 
     //---------------------------FREEZES X-POSITION---------------------------
 
     //Waits to ensure player doesn't move mid-air
     IEnumerator waitWhileJumping() {
         yield return new WaitForSeconds(1);
         playerSpeedX = 3;
     }
         
     //---------------------------CHANGING TIERS---------------------------
 
     IEnumerator moveLevelTier1() {
         if (grounded == true && positionTier1 == true) {
             yield return new WaitForSeconds (0.5f);
             transform.position += new Vector3 (0, 0, 1) * playerSpeedZ;
             positionTier1 = false;
             positionTier2 = true;
         }
     }
 
     IEnumerator moveLevelTier2() {
         if (grounded == true && positionTier2 == true) {
             yield return new WaitForSeconds (0.5f);
             transform.position += new Vector3 (0, 0, 1) * playerSpeedZ;
             positionTier2 = false;
             positionTier3 = true;
         }
     }
 
     IEnumerator moveLevelTier3() {
         if (grounded == true && positionTier3 == true) {
             yield return new WaitForSeconds (0.5f);
             transform.position += new Vector3 (0, 0, -1) * playerSpeedZ;
             positionTier3 = false;
             positionTier2 = true;
         }
     }
 
     IEnumerator moveLevelTier4() {
         if (grounded == true && positionTier2 == true) {
             yield return new WaitForSeconds (0.5f);
             transform.position += new Vector3 (0, 0, -1) * playerSpeedZ;
             positionTier2 = false;
             positionTier1 = true;
         }
     }
 }

My only issue is that the player jumps, then instantly moves along the z-axis. What I actually want to achieve is that it happens in one smooth motion, from jump to platform. I've been messing around with some loops and different functions but without luck.