It was certainly possible legislatively, and many WWII veterans were still serving in the military at the time or took it upon themselves to re-enlist voluntarily.
The Selective Training and Service Act of 1940 was passed on September 16, 1940, and was to be active until May 15, 1945, or a period of four years and eight months. On May 9, 1945, the act was extended for one year, until May 15, 1946. On May 14, 1946, the act was extended until July 1, 1946. On June 29, 1946, the provisions of the act were fixed by law to expire on either March 31 or July 1, 1947. Inductions into the military had been stopped by the director of the Selective Service on October 16, 1946, however. The draft was then not in effect for about two years.
On June 24, 1948, Congress passed "An Act to provide for the common defense by increasing the strength of the armed forces of the United States, including the reserve components thereof, and for other purposes," otherwise known as the Selective Service Act of 1948. The act was to remain in force for two years (until June 24, 1950).
The act required all men between the ages of 18 and 26 (born between 1922-1930) to register to be selected for potential military service, and men between the ages of 19 and 26 (born between 1922-1929) were liable to be inducted for a maximum of 21 months of active-duty military service. After, they were required to serve five years in a reserve component of the U.S. military.
The act specifically exempted former members of the United States military or members of the militaries of other countries allied with the United States during World War II (that is, prior to September 2, 1945) who had served between ninety days and twelve months on active duty between September 16, 1940 (the date of passage of the Selective Training and Service Act of 1940) and the date of passage of the act of 1948.
There were a few exceptions, however. Under the act, a man who fit the above categories could still be liable for induction after Congress declared a national emergency or war if;
(A) The local board determines that he is regularly enlisted or commissioned in any organized unit of a reserve component of the armed force (or the Coast Guard) in which he served, provided such unit is reasonably accessible to such person without unduly interrupting his normal pursuits and activities (including attendance at a college or university in which he is regularly enrolled), or in a reserve component (other than in an organized unit) of such armed force or the Coast Guard in any case in which enlistment or commission in an organized unit of a reserve component of such armed force or the Coast Guard is not available to him.
(B) The local board determines that enlistment or commission in a reserve component of such armed force or the Coast Guard is not available to him or that he has voluntarily enlisted or accepted appointment in an organized unit of a reserve component of an armed force other than the armed force in which he served or in the Coast Guard.
In addition, a person who completed at least three years of active-duty service and was honorably discharged after the date of passage of the act was exempt from service. On June 23, 1950, the Selective Service Act of 1948 was extended until July 9, 1950. The Korean War began on June 25, 1950. On June 30, 1950, the act was extended again, for a period of one year (to expire on July 9, 1951).
On June 19, 1951, the Universal Military Training and Service Act was passed, which lowered the ages of men liable for induction to 18 years and 6 months old and extended the active-duty service obligation to 24 months. This act also added one more year to the reserve obligation, to six years. This act did not have an expiration date, and, amended various times (1967, 1969, and 1972), formed the basis of conscription law in the United States until 1973.