I've searched through years 1955 to 1964, and did not find signs of Copyright Renewals for Volumes I, II, or III of Basore's translation of Seneca's Essays and letters. Below I have provided a google doc, in which I have typed the "theoretical" entries that would have preceded and succeeded a Basore entry for each year, so that my investigation can be verified/ corrected by others:
https://docs.google.com/document/d/1voHoZP-VNiiPU-F-tBcStP2JphLIqGSJnjGystmqfrE/edit?usp=sharing