Don't sell it short with "some kind of service". That rifle was undoubtedly in the AEF's hands by summer 1918 and saw some serious service. Remember Belleau Wood was summer 1918 and the Meuse-Argonne was a little later in the year. The war continued until 11/11/18 with action even on that very day.
In WWII it probably never left the states. It was most likely used at a camp gate, guarding a telephone or western union exchange building, guarding docks on either coast or one of the thousands of railroad bridges throughout the country. It was referb'd to serve in WWII and probably sold out of service after that.
If you want to learn a good deal about your rifles production I suggest you look up The Story of Eddystone on the Remington Society website. It's free and extremely informative.