The electron is always a quantum field. The quantum field can be regarded to describe a particle if and only if the field has a nonzero expectation only in a region small compared to the whole system considered. Thus we may say that the field is a particle as long as this condition is satisfied. Because of the dispersion of the field caused by the slits, this condition stops to be satisfied almost immediately after the field (with support large enough to cover both slits) passed the double slit. Thus it is no longer justified to talk about a particle.
The situation is similar as with a sphere of glass. If you throw it, you may regard it as a particle. But if it hits an obstacle and fragmentizes, it is no longer localized enogh to deserve the name of a particle.
The field passes the doulbe slit in a fashion similar as a water wave would do, except with quantum corrections.