This is a legal question more than a history question. If we're talking about the Constitution as interpreted today (assuming neither amendment existed), the answer is you probably don't an amendment to pass a modern equivalent of the Volstead Act.

But, at the time, the Commerce Clause (the justification for the Controlled Substances Act) was interpreted far more narrowly. While the Court was in a period of flux in the first two decades of the 20th Century, there was certainly a very realistic possibility that the Supreme Court would strike down such a law as unconstitutional. The big things to keep in mind is that it's the Interstate Commerce Clause. It deals primarily with buying and selling items across state lines. As time went on, the ability of the Federal government to regulate commercial activity within the state and non-commercial activity that affected commerce increased, but this wasn't cemented until United States v. Darby Lumber in 1941 and Wickard v. Filburn in 1942. ETA: I'll also add United States v. Wrightwood Dairy Co. in 1942 as well as a case that's often ignored but, together, make up pretty much a trinity for broad modern Commerce Clause jurisprudence.