Various thermal treatments have long been known to be effective in tailoring structural and magnetic properties of amorphous ferromagnetic wires, consequently, a desired magnetic anisotropy can be controlled for specific sensing applications. This paper deals with the effects of annealing and applied stresses on the magnetization processes and magnetoimpedance (MI) in Co71Fe5B11Si10Cr3 glass-coated microwires having amorphous or partially crystalline structure. The alloy under study has a small positive magnetostriction (approximately $10^{-8}$ ) in its amorphous unstressed state. By applying a tensile stress to amorphous microwires, an abrupt transformation of the hysteresis loop is observed owing to the anisotropy type change due to stress-dependent magnetostriction which changes sign. The modification in the anisotropy type greatly enhances the stress sensitivity of higher frequency harmonics induced during re-magnetization and MI. The wires with a partially crystalline structure did not exhibit a noticeable stress dependence of magnetic properties, but after annealing a significant increase in coercivity was observed after applying a tensile stress. The obtained results were investigated in terms of a magnetostrictive model of magnetic anisotropy offering a reasonable explanation.