A high-order elastic-viscous split stress/streamline-upwind Petrov-Galerkin (EVSS/SUPG) finite element method has been developed for the simulation of steady viscoelastic flows. Two benchmark problems an selected for the study: the flow of an upper-convected Maxwell fluid around a sphere in a tube and through a four-to-one axisymmetric smooth contraction. Our major concern is to verify p-convergence, i.e. whether solutions converge as the polynomial order of the approximation increases. The p-convergence is verified by means of a residual norm as well as by the stress profiles on the specific flow boundaries where the solutions are endowed with stress boundary layers. We have shown that for our SUPG scheme to be p-convergent, the upwind effect must vanish on stationary walls. Having discussed available SUPG schemes, we propose a SUPG formulation which, in the main flow region, has the same upwinding magnitude as earlier schemes, while it vanishes on stationary walls. Using this new EVSS/SUPG algorithm, we show p-convergence (1 less than or equal to p less than or equal to 7) up to a Weissenberg number We of 2.0 for the sphere problem and up to We=10.0 for the axisymmetric 4:1 contraction with a rounded re-entrant corner.