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Results on the giant magneto-impedance effect in thin as-cast glass-coated Co-based amorphous microwires and in such microwires after glass removal are reported. The results are explained by taking into account the complex magnetic structure of the region delimited by the magnetic penetration depth at high frequencies, which is responsible for the magnitude of the impedance variation. The interdomain wall between the inner core and the outer shell of the microwire domain structure is enclosed in this region, and it has a significant contribution to the magnitude of the impedance variation at various frequencies. Glass removal determines stress relief, which results in the expansion of the interdomain wall mostly to the detriment of the outer shell, which vanishes. The changes in the magneto-impedance effect with glass removal are fully understood only by taking into account the characteristics of the complex near-surface magnetic structure. The parameters of the magneto-impedance response of thin microwires are a consequence of the interplay among the magnetic penetration depth, the location and width of the interdomain wall, their changes with glass removal, and of the general characteristics of the domain structure.