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Relativistic laser wakefield acceleration is characterized by an unsurpassed accelerating gradient, which is very suitable for electron acceleration over short distances and could be a promising candidate for next-generation compact accelerators. However, using this technique for positron acceleration is still challenging because positively charged particles are naturally defocused in nonlinear wakefields. Here we propose and numerically demonstrate a scheme to accelerate an externally injected positron beam in a nonlinear laser wakefield in a regime where a tail wave is formed behind density cusps of the wakefield. This tail wave can provide a focusing force in addition to longitudinal acceleration for the positrons. Three-dimensional particle-in-cell simulations demonstrate that a trapping efficiency of positrons of nearly 100% in the nonlinear wakefield is possible. This scheme may open a simple way for compact positron acceleration to multi-100 MeV with terawatt-class laser systems at high repetition rates without the need for special laser modes and plasma structures.