Agent Spalko said:
Yeah, a plain white T-shirt is the uniform of detainment.
I was waiting for the Ford Finger and the line "You find that man!"
Men wore T-shirts in the '50s, army men. They started wearing them around WWII. Even men of the older generation began wearing T-Shirts. Besides, he's not wearing it the whole movie.
http://www.zoozoo2.com/t-shirt-history.html
''By World War II both the US Navy and army were wearing standard issue t-shirts as underwear. However, it was really in the 1940’s that the t-shirt really got going with returning US servicemen.''
''During W.W.I, American troops wore wool uniforms during hot summer days in Europe and noticed European soldiers wearing lightweight cotton undershirts. This cool apparel caught on fast with the Americans and by W.W.II, both the Army and the Navy included them in their uniforms.
Up until the 1950’s, t-shirts were still considered underwear, until John Wayne, Marlon Brando, and James Dean shocked Americans by wearing their “underwear” on T.V. In 1955, James Dean helped make the T-shirt a standard item of clothing in Rebel Without a Cause.''
http://northshoreshirts.com/tshirthistory.html
The T-shirt was used heavily by the army, who Indy is being captured by. Being that his clothes are no longer good because of radiation, they issue him underclothes. They're not going to issue him a full dress suit to wear, especially since when they first capture him they're suspecting him being a communist sympathizer.