Operating power amplifiers (PAs) at lower input back-off (IBO) levels is an effective way to improve PA efficiency, but often introduces severe nonlinear distortion that degrades transmission performance. Amplitude-phase-time block modulation (APTBM) has recently emerged as an effective solution to this problem. The intrinsic amplitude and phase constraints of each APTBM block can be leveraged to mitigate PA-induced nonlinear distortion via constraint-guided signal reconstruction. However, existing reconstruction methods apply these constraints only heuristically and statistically, limiting the achievable IBO reduction and PA efficiency improvement. This paper addresses this limitation by decomposing the nonlinear distortion into dominant and residual components, and accordingly develops a novel two-stage signal reconstruction algorithm consisting of coarse and fine reconstruction stages. The coarse reconstruction stage eliminates the dominant distortion by jointly exploiting the APTBM block structure and PA nonlinear characteristics. Subsequently, the fine reconstruction stage minimizes the residual distortion by casting it as a nonconvex optimization problem subject to explicit APTBM constraints, for which a closed-form solution is derived. The proposed algorithm is validated through comprehensive numerical simulations and testbed experiments. Results show that, without compromising transmission quality, the proposed algorithm enables an additional IBO reduction of approximately 5 dB in simulations and 2 dB in experiments over baseline methods, yielding relative PA efficiency improvements of 77.8\% and 30.9\%, respectively.