Laser-driven ion acceleration is theoretically/numerically mostly studied with the assumption of an idealised main ultrashort pulse of the Gaussian temporal shape, where nanosecond/multi-picosecond pedestals and short prepulses preceding the main pulse can be incorporated in the form of modifications in the initial density profile of irradiated ionised targets. This paper shows that the relatively slowly rising edge (also called picosecond ramp) of the main ultrashort pulse, usually neglected in previous studies, can substantially change the efficiency of the target normal sheath acceleration of ions depending on the laser intensity. The rising edge can enhance ion acceleration at mildly relativistic laser intensities, but increases the divergence and reduces the cutoff energy of accelerated ions at highly relativistic intensities relevant to petawatt lasers.