Finding the axis of a curving pipe

I am making a game similar F-Zero or Wipeout, where at times the player travels over the surface of a pipe, so the movement should always be relative to the pipe.

My prototype has been using the built in Cylinder primitive as the pipe, so I can get the long axis of the pipe by finding the Up vector of the cylinder the player is above. However, the goal is to have curved pipes, so I won't just be able to use the pipe's up vector to find the 'axis' of the pipe, which is curved. Does anyone have experience with this sort of issue?

I essentially need a better way of finding 'newforward' in the following code.

         RaycastHit hit;
         Physics.Raycast(controller.position, -controller.transform.up, out hit);
         Vector3 gravityDirection = hit.normal;
         Vector3 newforward = hit.collider.transform.up;
         
         Vector3 perp = Vector3.Cross(gravDirection, newforward).normalized;
         moveDirection = (perp * Input.GetAxisRaw("Horizontal")) * speed;
         moveDirection+= Input.GetAxisRaw("Vertical") * newforward * speed;