The extraction of accelerated positrons from the electron-beam-driven nonlinear plasma wakefield is investigated. If the trailing positron beam traverses a plasma density down-ramp at the exit of the plasma, it can become ring-shaped due to the defocusing force of the elongating wake bubble produced by the electron drive bunch. Ionization-induced head erosion process can be used to mitigate this effect because the drive electron beam leaves a hollow plasma channel with zero focusing force along the propagation axis. Particle-in-cell simulations are performed for the case where a drive-trailing electron bunch configuration traverses through a high-Z tantalum target placed at the plasma entrance, during which each bunch generates electron-positron pairs. If the plasma density upstream of this converter foil is correctly chosen such that the bubble length is equal to the inter-bunch spacing then the secondary positrons experience both a focusing and accelerating force and can gain energy from the wake. The simulation results demonstrate that when the uniform plasma length is longer than the beam head erosion distance for the drive electron bunch, the positrons can be extracted with a far smaller divergence than when they are extracted by the usual plasma density down-ramp. The transverse density distribution and the positron spectra of the trailing positrons for various acceptance angles of the spectrometer are presented to discuss the detectability of the accelerated positrons under experimental conditions at FACET-II.