You are actually correct -300 Savage was the initial start for the 308 -straight from Wikipedia. found some other articles that backed this up.
During the 1940s, the .300 Savage became the basis for experiments on behalf of the U.S. Military that resulted in the development of the T65 series of experimental cartridges. The original experimental case design by the Frankford Arsenal was designated the T65 and was similar to the .300 Savage case, but with less taper. The experimental cases were made from standard .30-06 Springfield cases which gave a little less capacity than standard .300 Savage cases because the Frankford Arsenal cases had slightly thicker case walls. The later T65 iterations were lengthened compared to the original T65 case and provided a ballistic performance roughly equal to the U.S. military .30-06 Springfield service cartridge. Over forty years of technical progress in the field of propellants allowed for similar service cartridge performance from a significantly shorter, smaller case with less case capacity.[2][3]
Winchester saw a market for a civilian model of the late T65 series designs and introduced it in 1952, two years prior to the NATO adoption of the T65E5 experimental cartridge iteration under the 7.62×51mm NATO designation in 1954. Winchester branded the cartridge and introduced it to the commercial hunting market as the .308 Winchester
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