A hard magnet biasing structure for a CPP-GMR or CPP-TMR read head for a magnetic recording disk drive is located between the two sensor shields and abutting the side edges of the sensor free layer. An insulating layer is located between the biasing structure and the lower shield and the side edges of the free layer. The biasing structure includes a seed layer of either Ir or Ru, a layer of ferromagnetic chemically-ordered FePt alloy hard bias layer on the seed layer, and a Ru or Ru/Ir capping layer on the FePt alloy hard bias layer. The FePt alloy has a face-centered-tetragonal structure with its c-axis generally in the plane of the layer. The relatively thin seed layer and capping layer allow the biasing structure to be made very thin while still permitting the FePt alloy hard bias layer to have high coercivity (Hc), a high remanent magnetization-thickness product (Mrt) and a high squareness (S=Mrt/Ms).