From the Remington history page:
Firearm Model History
Model 760 "Gamemaster"
Description: Slide action, high-power hunting rifle which replaced the Remington Model 141 and was later replaced by the Remington Model 7600.
Introduction Year: 1952
Year Discontinued: 1981
Total Production: Approximately 1.03 million
Designer/Inventor: L.R. Crittendon & William Gail Jr.
Action Type: Slide action
Caliber/Gauge: .30-06 Sprg. – 1952
.300 Savage – 1952-1960
.35 Rem. – 1952-1967
.270 Win. – 1953
.257 Roberts – 1954-1961
.244 Rem. – 1957-1960
.308 Win. – 1957
.222 Rem. – 1958-1960
.280 Rem. – 1958-1967
.223 Rem. – 1964-1968
.243 Win. – 1968
.6mm Rem. – 1968
.35 Rem. – 1980 only
Serial Number Blocks: 1951-1967 1,000-541,000
1968-1975 6,900,000-7,499,999
1975-1978 A6,900,000-A7,499,999
1978-1981 B6,900,000-B7,499,999
Various Models: 760A Standard – 1952
760B Special – 1952
760D Peerless – 1952
760F Premier – 1952
760ADL – 1954
760BDL – 1966
150th Anniversary Edition – 1966
760BDL Left-hand - 1966
American Bicentennial Edition 1976
760C – 1960 (carbine)
760CDL – 1960 – 1964 (carbine)
If I am reading this right the .308 was only offered in 1957 and the carbine in 1960 thru 1964. If that is right then the 308 carbine was not offered. MIKE.