TheNonuniform Magnetic Reconnectionmodel presented by Priest and Lee (1990) was the first to incorporate several features observed in previous numerical models, in particular the highly curved field in the inflow region, which relaxes one of the key assumptions of the classicalAlmost-Uniformreconnection model of Petschek (1964). Also present is a strong jet of plasma emerging from the reconnection region along the separatrix and a negative current spike at the outflow from the central diffusion region. In this paper we present a generalisation of the Priest-Lee model to include pressure gradients in the inflow region, so relaxing the other main assumption of the Petschek and Priest-Lee models, namely that the inflow is current-free or potential. This produces several reconnection regimes similar to those described by Priest and Forbes (1986) in theirUnified Almost-Uniformsolutions, where significant pressure gradients are included, but their inflow, like Petschek's, is almost-uniform with slightly curved field lines. We investigate the behaviour of the inflow magnetic field without linearising about a uniform field. We study both a shockless, incompressible outflow and one containing weak shocks, where the type of reconnection is dictated by both the inflow and outflow boundary conditions. In order to introduce a pressure gradient into the Priest-Lee model an extra non-potential field in thex-direction is added. This field has a uniform current and does not alter theY-point nature of the neutral points at the ends of the diffusion region but does alter the position of the separatrix, where the flow becomes singular, and hence the position of the Alfvénic discontinuity and the strength of the plasma jet. The maximum reconnection rate given by the model scales as (1 +c)3/2Rme−½, where the parametercis a measure of the current and is such thatc>0 gives fast- or slow-mode expansions andc< 0 gives slow-mode compressions. These regimes are also found in theUnified Almost-Uniformmodel. Fast reconnection with a reconnection rate in excess of the pure Sweet-Parker scaling is therefore possible in the flux pile-up regimes withc> 0. For a given large magnetic Reynolds number, a fast reconnection rate of, say, 0.1 times the Alfvén speed is possible providedcis large enough (e.g.c> 10 forRme= 105).