As far as I know, the reshuffling of the Senate wasn't a one-time deal in 1788, as new states entered the Union the Senate increased by two (in order to keep states' equality) the term for one of the new senators would be prorogued anyway, if they belong to different classes. Reshuffling, therefore, does not require a constitutional amendment.
Admitting Senators from new states does not prevent any existing senator from completing his/her term. This "reshuffle" does. Since the Constitution expressly gives each Senator a term of six years, altering the terms of any existing senators would require an Amendment.