The model of time-dependent Petschek-type reconnection is extended to take into account first-order corrections with respect to the reconnection rate. The plasma velocity inside the outflow region turns out to be smaller than the zero-order Alfvén velocity, and therefore strong reconnection is less effective in plasma acceleration. The model is also applied to study the effect of incoming waves on reconnection. We show that incoming waves of Alfvén type, carrying an electric current that is antiparallel to the initial current in the sheet, increase the reconnection rate and cause additional acceleration of the plasma, so that this wave can be considered as a potential trigger.