I don't know what a 1908 Winchester is.
Is that a patent date, maybe?
On the rifle or on the silencer?
What is the action type?
Can you post pictures?
Whatever the rifle, the Maxim Silencer came out in 1909 and was regularly available until 1934 when the NFA applied heavy restrictions and high taxes on them.
Maxim would sell you a complete gun + silencer outfit, thread your barrel to take their silencer, or sell a no-gunsmithing adapter.
If you are in the USA it is not legal to own a silencer without proper registration when new and payment of the transfer tax every time it changes hands.
I do not know if it is possible to legalize an old silencer that turns up in Grandpa's closet or some such.