In the present work, the influence of a minor Hf addition on the atomic structure, crystallization behavior, thermal stability and magnetic properties of a Co-based metallic glass was studied. Thermal analysis indicates that the thermal stability of new glassy ribbons microalloyed with 2.5 at.% Hf is notably enhanced through increasing the incubation time prior to devitrification and enlarging the width of the supercooled liquid region from 72 K to 96 K. Magnetic studies reveal that the new glass exhibits an excellent soft magnetic behavior, i.e., a very low coercivity of 0.26 A/m in the relaxed state, and a comparable saturation magnetization as the Hf-free ribbon. Structural relaxation of the Hf-containing alloy upon isothermal annealing below the glass transition temperature, Tg, was investigated by differential scanning calorimetry (DSC) and high-energy synchrotron X-ray diffraction (XRD). The evolution of the recovered enthalpy with annealing time can be expressed by the Kohlrausch-Williams-Watts (KWW) exponential function with a Kohlrausch exponent of 0.88, indicating a broad spectrum of relaxation times. Analysis of the reduced pair correlation functions, G(r), reveals volume shrinkage upon annealing caused by elimination of liquid-like sites including free-volume and anti-free-volume according to the shift in the positions of the G(r) maxima in the medium-range scale. The influence of structural relaxation on the variation of the Curie temperature and the coercivity of the new Hf-microalloyed glassy ribbon is discussed. A faster evolution of the Curie temperature with annealing time compared to the coercivity indicates a preferential dependence of the former on the chemical short-range order (SRO).