We propose a positron acceleration scheme in which a laser-driven positron beam is injected into a solid target hit by a laser beam and accelerated in the sheath field on the target back side. The positron beam injection and acceleration in the target have been investigated with numerical simulation. The feasibility of such an acceleration scheme was proved according to the simulation results, which show that a 10 MeV positron beam can be accelerated up to 30 MeV. The dependency of the positron beam properties on the positron injection location, injection time, and target thickness was studied. Related acceleration details were obtained and analyzed. The acceleration scheme provides a method in positron energy controlling and its related applications.