In international chess, there were very few women's tournaments held in the 1950s when Zvorykina was at her peak, but she tied for fourth place at the 1952 Moscow event and beat Anne Sunnucks (+1 =1 −0) in the USSR versus Great Britain Match of 1954.

The 1960 match coincided with her mother's terminal illness and this undoubtedly affected her play, resulting in a loss by a 4½–8½ margin.

In the 1960s, she competed in only a small number of international tournaments without much success; a new wave of strong players were by then in the ascendancy, most notably the Azerbaijani, Tatiana Zatulovskaya and the Georgians, Nona Gaprindashvili and Nana Alexandria.

In World Championship Candidates tournaments however, she remained a consistent and respected performer throughout the 1950s and well into the 1960s, never achieving less than a top 5 finish.

Representing the Soviet Union at the Women's Chess Olympiads of 1957 (Emmen) and 1963 (Split), Zvorykina produced two sparkling, medal-winning performances.

Zvorykina spent some time in Moscow, when her husband was appointed Head Coach there; later, she lived in Minsk, where she ran a chess school, although her career had previously been in engineering.