In this work, we carry out a systematic study on the evolution of relative intensity noise (RIN) in a high-power narrow-linewidth fiber laser system based on phase-modulated single-frequency seed laser. The experimental results clarify that enhancement of the RIN mainly occurs when the seed laser gets amplified from low power to medium power. In addition, further power scaling could lead to reduction of the RIN until it reaches saturation on the condition that the transverse mode instability (TMI) effect is effectively suppressed. Apart from the power scaling process, the phase-modulation would also lead to enhancement of the RIN at specific frequencies corresponded to the modulation frequency. More importantly, the contrast experiments reveal that the TMI effect could lead to obvious enhancement of the RIN of the signal laser during amplification well below the TMI threshold defined by the traditional method. Therefore, apart from degeneration of the RIN during amplification, potential enhancement of the RIN induced by the phase-modulation or the TMI effect below its traditional threshold should be carefully considered and controlled to achieve high-power low-noise fiber laser. Overall, our experimental results could provide a well reference for design of high-power low-noise fiber lasers and high-power narrow-linewidth fiber lasers.