OP - it might help to research some names - during WWI, three factories in USA made rifles chambered for 303 British - for the British military - Eddystone, Winchester and Remington - at least some of them went into the 6##,### serial number range - They are often called "P14" or "Pattern 14", and many people refer to them as "Enfield" rifles because they were initially designed at the Enfield Armoury in Britain.
The common WWII rifles used by Canada and others was either a No. 1 or a No. 4 "Lee Enfield" - is not the same rifle at all to an "Enfield" P14, but both used 303 British cartridges.
It might help your search a bit to clarify the name? But, as above - pretty much looking for a needle in a hay stack - but occasionally, odder things have occurred!!! Have to try!!