A read head, which has a head surface facing a moving magnetic medium, includes a read sensor that has first and second side top surface portions and a central top surface portion located between the first and second side top surface portions. First and second overlaying lead layers interface the first and second side top surface portions. First and second hard bias and tapered lead layers interface the first and second overlaying lead layers. A central top surface portion of the read sensor has a width that defines a track width of the read sensor. A method of making the read head includes ion-milling a partially oxidized portion of a cap layer and, after depositing the aforementioned first and second hard bias and tapered lead layers, preferentially reactive ion etching (RIE) the overlaying lead layer not covered by the first and second hard bias and tapered lead layers, so as to define the central top surface portion of the read sensor.