There has been a notable focus on the adoption of jamming-based technologies, which involve increasing the friction between grains, layers, or fibers to achieve variable stiffness capability in soft robots. Additionally, magnetorheological elastomers (MREs) that show magnetic-field-dependent viscoelasticity have great potential as a material for varying stiffness. This study proposes a hybrid method (magnetic jamming of MRE fibers) for enhancing the stiffness of soft robots, combining a jamming-based with a viscosity-based method. First, a fiber jamming structure is developed and integrated into a soft robot, the STIFF-FLOP manipulator, to prove the concept of the magnetic jamming of MRE fibers. Then, based on the proposed method, a variable stiffness device actuated by electro-permanent magnets is developed. The device is integrated into the same manipulator and the electronically controlled magnetic jamming and stiffening of the manipulator is demonstrated. The experimental results show that stiffness gain in bending and compression is achieved with the proposed method. The outcomes of this investigation demonstrate that the proposed hybrid stiffening technique presents a promising avenue for realizing variable and controllable stiffness in soft robots.