We present a scheme of fiber-connector-type carbon-nanotube-based saturable absorber (CNT-SA) with enhanced thermal damage threshold, in which the CNTs are deposited on the fiber connector end in a ring pattern for evanescent-field interaction instead of direct interaction. The thermal damage threshold of such CNT-SA is found to be increased by around 130% compared with an evenly deposited one. An all-fiber Fabry-Perot (FP) linear cavity passively mode-locked laser is further constructed incorporating the prepared CNT-SA, where the optical power is confined in a relatively short laser cavity to investigate the thermal damage threshold and the performance of the CNT-SA. Stable output pulses with a fundamental repetition rate of 211.84 MHz and a pulse width of 680 fs are generated from the fiber laser. The mode-locking operation can be maintained an intra-cavity average power of 30 mW, indicating that the CNT-SA can withstand a relatively high optical power without performance degradation.