5.56 calibre didn’t even exist in 1963. The Manlicher-Carcano rifle Oswald used was 6.5x52mm so it was already a smaller calibre than the usual rifle round.
Throughout 1962 and 1963, the U.S. military extensively tested the AR-15. Positive evaluations emphasized its lightness, “lethality”, and reliability.[17] However, the Army Materiel Command criticized its inaccuracy at longer ranges and lack of penetrating power at higher ranges.[43][37][17] In early 1963, the U.S. Special Forces asked, and was given permission, to make the AR-15 its standard weapon.
The Secret service was using this “new rifle”. In 5.56.
It’s civilian equivalent, .223 Remington certainly did. And the idea that the 6.5 carcano, especially using it’s standard projectile, is a SCHV cartridge, is completely wrong.