Optical frequency combs based on fiber lasers mode-locked (ML) with a nonlinear amplifying loop mirror (NALM) have become the backbone of many cutting-edge applications, ranging from precision spectroscopy to quantum physics. Being extremely precise measurement tools, understanding their passive stability and low-noise operation regimes is vital. While several influences on the laser noise have been studied, many parameters remain poorly understood. Here, we systematically analyze under which preconditions the artificial saturable absorber settings of the laser can be modified during operation without losing mode-locking and the effects on laser noise, the spectrum and the output power. Our results show that it is possible to decrease the amplitude noise (AM noise) of the laser by more than 50 % by simply rotating a wave plate within the laser cavity. Additionally, we discuss differences to a similar effect observed in a NALM-alike laser amplifier and of changing the output coupling. These findings deepen our understanding and capabilities of optimizing the noise performance of ML fiber lasers, enable us to investigate new parameter spaces, and can be used to further optimize the noise performance of the NALM laser design, making it an ideal light source for advanced setups both in research and industry.