I think the proper way of stating it is that Savage didn't INTRODUCE the .308 family of cartridges until 1957, as in, their 1957 catalog year.
That means that they would have been manufacturing the guns in 1956 for shipment in the 1957 catalog year.
I don't have any catalogs or advertisements from that time frame, but it's very likely that Savage would have announced the new cartridges sometime in the second half of 1956, much like companies sometimes still do.
So yeah, I'm not surprised at all that you have a 99F in .308 with a 1956 date code on it.