"The "UMC" on the label stands for Union Metallic Cartridge Co. and as I recall that name disapeared from the Remington labels around 1910 or so."
Sorry, but that is incorrect.
Remington and UMC became part of the same organization (under the umbrella of Marcus Hartley and Partners) in 1888.
Remington and UMC were consolidated as a single entity in 1912. Prior to that time Remington ammunition and UMC ammunition were produced in separate factories under their own brandnames.
After that, the UMC appeared, disappeared, appeared, disappeared, was used on some product lines but not others, etc., for decades. Often, even when the UMC logo wasn't prominently displayed on the box, cartridges would be headstamped REM-UMC.
It was used as an advertising mark, and remains in use to this day.
The TRUE way to tell the age of that tin?
One word...
DuPont.
DuPont purchased Remington in 1933.
The DuPont logo would not have appeared on any Remington products prior to that time.
My guess, your tin was made sometime between 1933/1934 and the start of World War II.