Klein's experience as a performer increased throughout the 1930s. Nor was he idle as a composer. An encounter with a leading contemporary composer, then visiting Prague, inspired him to complete one of his first compositions extant from that decade. In 1933, Darius Milhaud, a member of the so-called Les Six group of French composers, visited Prague at the invitation of the ISCM (International Society for Contemporary Music), still today the foremost international forum for the promotion of new music; its Czech chapter was then especially active. Klein's Little Suite for piano, dated 27 April 1933, is his only surviving composition since the Suite lyrique of almost four years earlier – at least, there is no evidence to suggest that he composed anything between 1929 and 1933; his intense piano studies might have been the reason for his compositional silence.

The Little Suite comprises two short movements, an Andante followed by a Vivace, with an inscription at the end, ‘After Milhaud's concert’. In all likelihood, judging from the notation of the first movement in particular, and the single date it bears, the piece was written quickly, in a flash of Milhaud-inspired creativity. In the first movement, Klein is clearly experimenting with the type of jazz-influenced harmony that Milhaud uses. The second movement is a rapid nine bars, which tries to capture the velocity and unbounded energy of pieces such as Milhaud's 1923 jazz-like ballet score La Création du monde (‘The Creation of the World’), or his earlier ballet Le boeuf sur le toit (‘The Ox on the Roof ’). Although there seems to be no record of what he might have heard at that 1933 concert, this modern, Gallic style was something which clearly impressed him, and was championed by sections of the Czech avant-garde generally.

Apart from attending classes in general musicianship as a part-time student at the Conservatoire, Klein was not receiving any formal composition tuition at this time, and would not do so until his studies with Alois Hába began in 1939. Consequently, a work such as the Little Suite, not especially remarkable apart from the age of its creator, is unencumbered by compositional theory.