Applying a two-dimensional, compressible Hall magnetohydrodynamic (MHD) model, we study reconnection of magnetic fields in various cases: symmetric and non-symmetric antiparallel magnetic fields, and also skewed magnetic fields. The magnetic reconnection process is initiated by switching on a localized resistivity assumed to be a Gaussian function of spatial coordinates. As initial condition we set a one-dimensional steady-state current sheet. The obtained numerical solution indicates that reconnection process evolves asymptotically to the stationary Petschek-type regime in the case of inhomogeneous resistivity. The internal reconnection rate is found to be proportional to square root of the inverse local magnetic Reynolds number. The external reconnection rate is found by matching the external Petschek solution and the internal diffusion region solution for various cases of steady-state two-dimensional reconnection in a compressible plasma. The obtained general formula for the reconnection rate yield those of Sweet–Parker or Petschek in particular cases of pure homogeneous or strongly localized resistivity. In case of skewed reconnecting magnetic field, the reconnection rate is proportional to the sine of angle between magnetic field and reconnection line. The Hall parameter is found to be responsible for generation of the so called Alfvén wings structure. These wings are related to the Hall MHD Alfvén waves propagating faster than the usual Alfvén waves in the ideal MHD model. The obtained Alfvén wings are characterized by intensive field-aligned currents and large variations of the out-of-plane magnetic field and velocity components.