In this paper, we continue our previous study towards understanding the twodimensional hydrodynamic systems describing Oldroyd type incompressible viscoelastic fluids. We will decompose the deformation tensor into the strain and rotation components and look at their distinct contributions and structures in the small strain (with respect to viscosity) dynamics. In particular, we prove that there exist classical solutions globally in time if the strain component of the initial deformation is

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... enough, while we require no assumptions on smallness of the magnitude of the rotation component.