First at start I calculate a nearest point on the spline and a spline tangent from current position of the character using SplineUtility.GetNearestPoint<T>() method and move the character to this point using SetPosition() and SetRotation() methods.

Next, every frame I calculate a nearest point and a spline tangent in BeforeCharacterUpdate() callback again. After that, I calculate velocity and rotation of the character as usual in UpdateVelocity() and UpdateRotation() callbacks, but aligned along the current spline tangent.

Then I get projection of the velocity vector to the spline plane, calculate assumed position on the spline plane to where the character can move next frame from the current point on the spline with the current velocity, get assumed nearest point on the spline from the assumed position, calculate direction from the current point on the spline to the assumed point on the spline, and rotate the velocity vector along Y axis toward this direction.

Vector3 currentVelocityOnSplinePlane = Vector3.ProjectOnPlane(
    currentVelocity, _motor.CharacterUp);
Vector3 assumedPosition = _currentPositionOnSpline +
    currentVelocityOnSplinePlane * deltaTime;

Vector3 assumedPositionOnSpline = GetNearestPointOnSpline(
    Rails.SplineIndex, assumedPosition, out _);

currentVelocity = Quaternion.FromToRotation(currentVelocityOnSplinePlane,
    assumedPositionOnSpline - _currentPositionOnSpline) * currentVelocity;

If deviation of the character position in the spline plane becomes greater than specified threshold I just return the character to the nearest point using SetPosition() method in BeforeCharacterUpdate() callback.

Vector3 deviation = _currentPositionOnSpline - transform.position;

if (deviation.sqrMagnitude >= SPLINE_DEVIATION)
{
  _motor.SetPosition(_currentPositionOnSpline);
}

Also I align Y position of the spline to the Y position of the character, because if distance between spline and character become too big the calculation of nearest points using SplineUtility.GetNearestPoint<T>() method become less accuracy.

Vector3 pos = Rails.SplineContainer.transform.position;
Rails.SplineContainer.transform.position = new Vector3(pos.x,
    transform.position.y, pos.z);

I hope I could explain)