how to continue spline?

I've created a simple script that can go from A to B. But now I'm trying to have the ability to jump in the middle of a spline and continue to point B. I think my math is just wrong, but I really don't know. at one point I tried to set the parameter(progress).

 private void Update()
     {
        WalkLine();
 
         RaycastHit hit;
 
         Vector3 fwd = transform.TransformDirection(Vector3.down) * .5f;
 
         if (Physics.Raycast(transform.position, fwd, out hit, .5f) && hit.transform.tag == "Line" && railSet == false)
         {
             line = hit.collider.GetComponent<Line>();
             position = this.transform.position;
             railSet = true;
 
 
             Debug.DrawRay(transform.position, fwd, Color.red);
         }
         else
         {
             Debug.DrawRay(transform.position, fwd, Color.cyan);
         }
 
         print("position: " + position);
 
 
     }
 
     private void WalkLine()
     {
         if (railSet == true)
         {
 
             progress = GetTouchedPoint(position);
             transform.localPosition = line.GetPoint(progress);
 
             if (progress == 1 || cube.isGrounded == false)
             {
                 line = null;
                 railSet = false;
             }
         }
     }
 
 
     //this should be the point intersected by the ray and the collider
     public float GetTouchedPoint(Vector3 p)
     {
         Vector3 M = line.points[1] - line.points[0];
 
         print(M+ " * ( " + p + " - " + line.points[0] + ") / (" + M + " * " + M+") == "+ Mathf.Abs(Vector3.Dot(M, p - line.points[0]) /
             Vector3.Dot(M, M)));
         return
             Mathf.Abs(Vector3.Dot(M, p - line.points[0]) /
             Vector3.Dot(M, M));
     }