We present preliminary results that were collected using a new positron beam system at the University of Texas at Arlington. The beam allows for the coincident measurement of the energy of positron annihilation gamma photon and time-coincident positron-induced electrons. This system enables the parallel collection of a 2dimensional array of coincident pairs of gamma and electron energies over the full energy range of relevance. Here, we present gamma spectra constructed from the correlation of secondary electrons with the delayed annihilation of positronium, and the observation of pick-off annihilations that results from this analysis. These measurements will play an important role in determining the contributions due to pick-off annihilations to the Doppler-broadened gamma spectra from the internal or external exposed surfaces of materials. Subtracting the background properly is crucial for the measurement of the Doppler broadened spectrum of annihilation gamma photons originating from the annihilation of positrons with electrons of specific electronic core levels.