We report on the high-power amplification of a 1064 nm linearly polarized laser in an all-fiber polarization-maintained master oscillator power amplifier, which can operate at an output power level of 1.3 kW. The beam quality (M2) was measured to be <1.2 at full power operation. The polarization extinction rate of the fiber amplifier was measured to be above 94% before mode instabilities (MIs) set in, which reduced to about 90% after the onset of MI. The power scaling capability of strategies for suppressing MI is analyzed based on a semianalytical model, the theoretical results of which agree with the experimental results. It shows that mitigating MI by coiling the gain fiber is an effective and practical method in standard double-cladding large mode area fiber, and, by tight coiling of the gain fiber to the radius of 5.5 cm, the MI threshold can be increased to three times higher than that without coiling or loose coiling. Experimental studies have been carried out to verify the idea, which has proved that MI was suppressed successfully in the amplifier by proper coiling.