This paper presents the results of numerical simulations of a crystalline undulator based on the channeling of 20 GeV electrons and positrons. The device considered is characterized by a small amplitude and a short period of periodic bending. Calculations have been performed accounting for all-atom interactions using the MBN Explorer software package. The effect of low crystal thickness (less than a channeling oscillations period) on radiation spectrum was studied. A new scheme to product a high-energy radiation was proposed. It is based on the short-period small-amplitude crystalline undulator and allows decreasing the intensity of the non-undulator part of the spectrum.