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Laser–plasma accelerators (LPAs) have been demonstrated as one of the candidates for traditional accelerators and have attracted increasing attention due to their compact size, high acceleration gradients, low cost, etc. However, LPAs for positrons still face many challenges, such as the beam divergence controlling, large energy spread, and complicated plasma backgrounds. Here, we propose a possible multistage positron acceleration scheme for high energy positron beam acceleration and propagation. It is driven by the strong coherent THz radiation generated when an injected electron ring beam passes through one or more solid targets. Multidimensional particle-in-cell simulations demonstrated that each acceleration stage is able to provide nearly 200 MeV energy gain for the positrons. Meanwhile, the positron beam energy spread can be controlled within 2%, and the beam emittance can be maintained during the beam acceleration and propagation. This may attract one’s interests in potential experiments on both large laser facilities and a traditional accelerator together with a laser system.