We have succeeded in measuring electric-field versus current-density (E-J) characteristics of REBCO coated conductors under external magnetic field over wide-range electric-field window from 10(-2) down to 10(-11) V/m continuously by combining both transport- and magnetization-relaxation-measurements. High-temperature superconductors (HTSs) can cover broad aspects of applications for AC and DC mode operation such as superconducting motors and persistent-mode MRI magnets. Electric field induced in these applications vary significantly from 10(-2) down to 10(-11) V/m depending on the operating frequency. This indicates that the corresponding critical current density (J(c)) also varies due to the rounded E-J characteristics of HTS. Namely, it is inevitable to clarify the E-J characteristics in wide range of electric fields. In general, E-J characteristics are measured by current transport method or magnetization relaxation method. The standard transport measurement can cover typically from 10(-2) down to around 10(-5) to 10(-4) V/m because of the limit of voltage noise, whereas in a DC magnetization measurement such that using SQUID magnetometer for example, the induced electric field during the measurement is around 10(-8) V/m or less. In this study, we adopted Hall probe magnetic microscopy to expand the electric-field window in the measurements on E-J characteristics based on magnetization relaxation. We developed zero-dimensional fixed-point measurement with a time resolution less than 0.1 s to measure the initial decay of the magnetization, which extends the electric-field window up to similar level to that of transport measurement, i.e., at around 10(-5) V/m. It was followed by a one-dimensional line scan in width-direction for the electric-field range between 10(-8) to 10(-11) V/m. Furthermore, "flux annealing" method was adopted to access the electric-field range around 10(-11) V/m with shorter measurement time.