The intrinsic constraints in the amplitude of the accelerating fields sustainable by radio-frequency accelerators demand for the pursuit of alternative and more compact acceleration schemes. Among these, plasma-based accelerators are arguably the most promising, thanks to the high-accelerating fields they can sustain, greatly exceeding the GeV/m. While plasma-based acceleration of electrons is now sufficiently mature for systematic studies in this direction, positron acceleration is still at its infancy, with limited projects currently undergoing to provide a viable test facility for further experiments. In this article, we study the feasibility of using a recently demonstrated laser-driven configuration as a relatively compact and inexpensive source of high-quality ultra-relativistic positrons for laser-driven and particle-driven plasma wakefield acceleration studies. Monte-Carlo simulations show that near-term high-intensity laser facilities can produce positron beams with high-current, femtosecond-scale duration, and sufficiently low normalised emittance at energies in the GeV range to be injected in further acceleration stages.