Soft continuum robots have been widely used as guide wires or catheters for minimally invasive surgery (MIS), and the miniaturization and dexterity are very important characteristics of soft continuum robots. As a representative actuation method for soft continuum robots, the steering using an external magnetic field has been actively studied. The magnetic actuation method is appropriate for the miniaturization of soft continuum robots but shows limitations in implementing high dexterity. To achieve miniaturization and high dexterity, this paper proposes a magnetically steerable asymmetric magnetized soft continuum robot (AMSCR). The proposed AMSCR includes an active steering part (ASP) of a thin PDMS cylinder in which NdFeB powder is uniformly dispersed and magnetized asymmetrically to the longitudinal direction. Therefore, compared to the conventional symmetric magnetized soft continuum robot (SMSCR), the proposed AMSCR has a larger steering range and high dexterity. Through various experiments, simulations, and phantom experiments using the proposed AMSCR, we analyzed its characteristics, verified its enhanced steering range, and demonstrated the applicability to the surgeries requiring high dexterity in the eye or cerebrovascular.