Ultrafast microchannel-plate photomultipliers offer picosecond resolution in time-correlated photon counting, provided all additional causes of timing jitter be minimized. Here we analyze the contribution of the electronic circuit noise, mainly dependent on the fast preamplifier and no more negligible at picosecond level. Criteria are derived for minimizing this contribution by proper design or selection of the amplifier. We highlight that it does not make sense to select amplifiers with bandwidths of 3 GHz or more, looking mainly to the rise time of the fast microchannel-plate pulse and paying minor attention to the noise. We demonstrate that (i) the noise spectral amplitude has chief importance, (ii) the bandwidth providing minimum jitter is around 1 GHz, and (iii) satisfactory results are obtained with lower bandwidths, in some cases down to about 500 MHz.