There *were* millions of men in the civilian workforce with combat experience and training and still of age for service, and you could draft many back into the ranks - but obviously, that can't be done overnight.
Not to mention combat and discipline skills are highly perishable without routine exercise, which is why reservists are required to come in on weekends to train, something I doubt the millions of men who returned to a purely civilian life had been receiving over the past three years. Many of these men will need remedial training. It'll be faster then starting with scratch recruits, since these guys will be relearning skills they once knew, but it'll still take more time then if they had been regularly put through their paces each weekend like Soviet reservists prior to 1967 were.