It is well known that the classical one-pass node-to-segment algorithms for the enforcement of contact constraints fail the contact patch test. This implies that solution errors may be introduced at the contacting surfaces, and these errors do not necessarily decrease with mesh refinement. The previous research has mainly focused on the Lagrange multiplier method, but the situation is even worse with the penalty method. In a recent study, the authors proposed a modified one-pass node-to-segment algorithm which is able to pass the contact patch test also in conjunction with the penalty method. In a general situation, the pressure distribution transferred across a contact surface is non-uniform. Hence, even for a contact element which passes the contact patch test under a uniform distribution of the contact pressures, the transfer of a non-uniform state of stress may give rise to disturbances related to the discretization, which affect the accuracy of the analysis. This paper, following up to the previous study, develops an enhanced node-to-segment formulation able to pass a modified version of the contact patch test whereby a linear distribution of pressures has to be transmitted across the contact surface. The proposed formulation is illustrated and some numerical examples demonstrate the good patch test performance of the enhanced contact element.