This article considers transient, planar Poiseuille flows for viscoelastic fluids. We propose a novel time-dependent hybrid finite volume (fv)/finite element (fe) algorithm. This approach combines a Taylor-Galerkin fe-treatment for mass and momentum conservation equations, with a cell-vertex fv-discretisation of the hyperbolic stress constitutive equation. A consistent formulation for the constitutive equation is key. This incorporates fe and fv-treatment of the various terms. In this manner, an accurate transient algorithm emerges, which reproduces analytical solution structure, both in core-flow and across shear-boundary zones.