To address the challenges raised by the data management of exascale scientific data, error-bounded lossy compression has been proposed and well-researched as a prominent solution. Among the existing works, a recent trend leverages wavelet transforms in the error-bounded lossy compression task to effectively capture long-term data correlations within the inputs. Applying those transforms as data preprocessors and decorrelators, wavelet-based lossy compressors have achieved optimized compression rate-distortion on several datasets. However, certain significant limitations of wavelet-based compressors have also been observed: On one hand, attributed to the high computational cost of wavelet transforms, wavelet-based compressors suffer from relatively low computational efficiencies compared to other state-of-the-art compressors. On the other hand, one certain type of wavelet transform cannot perform well on all variations of scientific data. Consequently, to further fine-tune the wavelet-based scientific data lossy compression, more in-depth and systematic research and analysis needs to be conducted. In this paper, based on the FAZ auto-tuning-based modular compression framework, we have integrated a great number of wavelet transforms into the framework and evaluated them with various real-world scientific datasets and fields. From the analysis of those evaluations and the comparison to existing state-of-the-art wavelet-based and non-wavelet-based error-bounded lossy compressors, we conclude and present several essential takeaways for designing and optimizing the wavelet-based scientific error-bounded lossy compressor.