I was hopeing someone else would answer this because I am certainly no expert on this subject but I’m gonna give it a shot because as a beginner myself, I know how frustrating it can be when questions go unanswered. Hopefully some of the really knowledgeable folks will see this and give us both some education.
In Journal #469 Dick Fraser did an excellent article on interpreting Remington ammo box date codes. Some of his illustrations show early UMC boxes because Remington absorbed UMC, so to speak. His earliest box, which is nearly identical to yours, is listed as Pre-1911 but it has the red ball with UMC logo while yours has an oval with U.M.C. and Trade Mark on the left side. So I am speculating that your box goes back to around 1900, before or after a bit.
Hopefully someone who really knows will pipe in so don’t be discouraged. There are some really smart folks with tons of knowlege on here!