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We have developed a compact vibrating-sample magnetometer that uses a high-temperature superconductor superconducting quantum interference device (high-Tc SQUID) and a normal conductive pickup coil. To further increase the sensitivity of the developed system, we consider a technique to detect harmonic components induced at the pickup coil. We optimized the geometry of the pickup coil and sample to maximize the harmonic signals and compared the harmonic signals induced from different shapes of coils using simulation and experimental methods. The simulation and the experimental results agreed well, indicating that the optimized geometry of the pickup coil improves the induced harmonic components. Improvement in sensitivity with reduced mechanical noise can be attained using this detection technique.