In this study, it is demonstrated that the giant self-biased magnetoelectric (SME) response can be achieved from a center-clamped magnetostrictive–piezoelectric laminate composite by employing magnetic tip masses. An asymmetric laminate structure consisting of two different magnetostrictive layers (Metglas and nickel) with opposite signs of piezomagnetic coefficient is introduced to promote structural bending resonance, and the effect of layout change of attaching the magnetic tip masses on SME responses is systematically investigated. The highest SME effect is observed when all magnetic tip masses are loaded on the Metglas layer and their magnetization directions are normal to the Metglas surface. It is proposed that not only the parallel magnetic domains to external magnetic field but also the non-parallel magnetic domains effectively contribute to the total magnetostriction. The fabricated SME laminates exhibit giant SME voltage coefficients ranging from 14.11 to 52.35 V cm−1 Oe−1, depending on the direction of the fields of the tip magnets. These high SME voltage output values and their controllability are promising for precision field sensors, magnetic energy harvesters and field-tunable devices.