A recent paper by Hinch (J. Non-Newtonian Fluid Mech., 50 (1993) 161) has examined similarity solutions near a re-entrant corner for the UCM fluid. Hinch obtains a stress behavior like r-(2/3), but a Stream function which behaves like theta(7/3) near the wall. He concludes that no similarity solutions with the expected behavior of theta(2) exist. In this note, we remedy this defect by using a different behavior of the stream function near the wall. A matched asymptotic solution is constructed where the Hinch solution is the outer solution. The inner solution is governed by a nonlinear system of ordinary differential equations. By numerical integration, we find a solution at the upstream wall which matches up to the Hinch solution. At leading order, this matched solution satisfies the equations of motion in both the core and wall region. The solution near the downstream wall is left as a future problem.