By operating an intracavity semiconductor-optical-amplifier- (SOA-) based high-pass filter at the nearly transparent current condition, the supermode noise (SMN), the relaxation oscillation, and the single-sideband (SSB) phase noise can be simultaneously suppressed in an actively mode-locked erbium-doped fiber laser (EDFL). The SOA at the nearly transparent condition enhances the SMN suppression ratio of the EDFL from 32 to 76 dB at the cost of the phase noise degrading from -114 to -104.2 dBc/Hz and broadening the pulse width from 36 to 61 ps. With an optical bandpass filter, the SSB phase noise and the SMN suppression ratio can be further improved to -110 dBc/Hz and 81 dB, respectively. The EDFL pulse can be further shortened to 3.1 ps with a time-bandwidth product of 0.63 after compression.