I believe the first use of totally free floating barrels was in the early to mid 1930's. Competitors using M1903's chambered for the .30-06 round and using the new 172-gr. FMJBT arsenal match bullet as well as Western's 180-gr. FMJBT match bullet began noticing that the use of a sling in different ways could cause accuracy problems with the best lots of ammo. Thick beavertail fore ends were not common then and the standard ones did bend enough to touch the barrel. NRA rules prior to 1935 allowed only the M1903 but were then changed to any .30-06 rifle weighing less than 9 pounds. The Exec. Committee added that any rifle and any cartridge could be used in some long range matches. So, someone hogged out the barrel channel and accuracy got better. He told a few people who each told a few more and the improvement snowballed. (Or somthing like all of this)
Note: the above was told to me in 1967 by the old postman delivering mail to our house in San Diego. I was working in the opened garage routing out a stock to epoxy bed a Win. 70 barreled action and he came in asking me what I was doing. I explained and he got real excited and wanted to learn more. When I finished, he pulled out his billfold, opened it to a picture of several people holding M1903's wearing campaign hats then he pointed to one saying "That's me." He said that was a picture of the USMC Rifle Team at the Nationals in 1937, the year he legged out and got his Distinguished Rifleman's gold badge. He asked me if the US Navy was still using matchbook covers to tighten up their Garands then I brought out my 7.62 NATO M1, took the barrel group out and showed him the grey plastic steel; he was really excited to see how things were done some 30 years after he shot rifle matches with the M1903.
When Winchester introduced their Model 70 in 1937, there were three competition versions; Target Rifle and National Match in .30-06 with standard stocks and the Bull Gun in .300 H&H Magnum with a target stock having a wider beavertail fore end. Folks buying them usually ensured the barrel was well clear of the wood as well as removing the screw that went up through the standard stock's fore end to a dovetail socket on the barrel's bottom. Sometimes they would shim the receiver (match book covers were popular and they carried over to Garands some years later; this was pre-epoxy bedding times) in the stock to tighten its fit and keep the barrel "floating" in the fore end.
