A typical axially compact, square window core steering magnet has a very small 2D good field region (GFR) of 18% of the iron core aperture. It is weak with 85% of its dipole in the fringes. Increasing the currentfed area (coil) to increase the dipole strength lowers the good field zone. Some novel and simple coil shapes can expand their 2D/3D GFR to 94/40% of the iron aperture. They are axially compact, i.e., stronger with 60–65% of the dipole in fringes. The reduction in error sextupole field is more profound. A novel 2D coil shape with vanished 2D sextupole proves the concept. For axially short cores, a gap between the coil and the iron aperture can increase the 3D sextupole, missing in 2D simulations. A new asymmetric one-axis steering magnet core and coil shape with high dipole field and homogeneity is also given. The new, practical coil shapes significantly improve the dipole field homogeneity and strength of these magnets.