In this study we report on some critical factors affecting the sensitivity of a magnetic field sensor whose working principle is based on a linear DC field dependence of the second harmonic of the AC response in polycrystalline type-II superconductors. DC-fields down to 2 nT in magnitude could be detected by finding optimal conditions. The optimal sensor design was determined by studying superconductor cores having different geometries and coil configurations. The optimal AC-field frequency, which is required for excitation of the specimen, was found to be 50 kHz. The second harmonic signal changes quite linearly with the DC-field up to 0.1 Oe.