Reconstruction of standard-dose Positron Emission Tomography is vital for clinical diagnosis, while recent diffusion-denoising probabilistic models offer strong generative capabilities, when applied to this task, they often struggle with fine detail recovery, slow inference, and inadequate cross-slice continuity in 3D volumes. To overcome these issues, we introduce WiD-PET, which employs a wavelet transform to produce smaller wavelet-transformed inputs, and thereby reduces inference time to 10% of that required by the DDPM model. Additionally, a high-frequency enhancer is adopted for reconstructing fine and rich image details. Moreover, a spatial consistency feature extractor and spatial consistency attention are implemented to enhance cross-slice continuity in 3D PET reconstructions. Evaluations across dose levels (1/20, 1/50, and 1/100) reveal that WiD-PET consistently achieves superior reconstruction quality, detail preservation, and inference efficiency. Project page: https://github.com/SwingM/WiD.git .