Prospects for average power scaling of sub-MW output peak power picosecond fiber lasers by utilization of a Yb-doped tapered fiber at the final amplification stage were studied. In this paper, it was shown experimentally that a tapered fiber allows the achievement of an average power level of 150 W (limited by the available pump power) with a peak power of 0.74 MW for 22 ps pulses with no signs of transverse mode instability. Measurements of the mode content using the S2 technique showed a negligible level of high order modes (less than 0.3%) in the output radiation even for the maximum output power level. Our reliability tests predict no thermal issues during long-term operation (105 hours) of the developed tapered fiber laser up to kilowatt output average power levels.