A model is proposed capable of accounting for the local electric field increase in front of the lightning stepped leader up to magnitudes allowing front electrons to overcome the runaway energy threshold and thus to initiate relativistic runaway electron avalanches capable of generating X-ray and ray bursts observed in negative lightning leader. The model is based on an idea that an ionization wave, propagating in a preionized channel, is being focused, such that its front remains narrow and the front electric field is being enhanced. It is proposed that when a space leader segment, formed ahead of a negative lightning leader, connects to the leader, the electric potential of the leader is transferred through the space leader in an ionizing wave that continues into the partly ionized channels of preexisting streamers of the space leader. It is shown with numerical simulations that the ionization channels of streamers limit the lateral expansion of the ionization wave, thereby enhancing the peak electric field to values allowing an acceleration of low-energy electrons into the runaway regime where electrons efficiently generate bremsstrahlung. The results suggest that the inhomogeneous ionization environment at the new leader tip amplifies the production rate of energetic electrons relative to a homogeneous environment considered in the past studies.