In the past two decades, index-near-zero (INZ) modes and materials, with their spatial phase invariance and super coupling, gained increasing attention for applications in all-optical/quantum computing and communication. However, the modulation of INZ modes is typically complex and discontinuous, often achieved through intricate experimental methods, thereby hindering their widespread application. Here, we propose two deep-subwavelength magneto-optical one-way waveguides and discover three broadband tunable INZ modes, exhibiting predict able behavior dependent on the external magnetic field (EMF). By utilizing these INZ modes, we design broadband tunable all-optical phase modulators through straightforward EMF control. The tunable and predictable nature of INZ modes, combined with deep-subwavelength phase modulators, may advance miniaturized all-optical communication and computation.