A family of models for fast steady-state reconnection has recently been presented by Priest and Forbes, of which the Petschek-like and Sonnerup-like solutions are special cases. This essentially linear treatment involves expanding about a uniform flow and field in powers of the external Alfvén Mach number Me, and hence is valid for small values of that parameter. To lowest order, the discrete slow-mode compressions attached to the diffusion region are straight, while downstream of them the plasma flows at simply the external Alfvén speed vAe and the field lines are straight. Here we present an extension of these solutions to the next order, which not only reveals that the wave itself is curved (as are the downstream magnetic field lines), but also that the downstream solution is sensitive to changes in the upstream boundary conditions. In the downstream solution there is a free parameter, which may be specified as a downstream boundary condition. Thus the boundary conditions at both the inflow and the outflow boundaries are crucial in determining the nature of the reconnection.