I'm working on a new picture of my 1903A4, Gary.
You guys will like it, trust me.
But there's something wrong with those scenes in
Saving Private Ryan, regardless.
The U.S. Army used the 1903A4 Remington, with a 2-1/4x Weaver 330C or M73B1 scope. It wasn't really easy to remove the scope in the field, at least one windage screw on the Redfield Jr. base had to be removed. This allowed the front scope ring to pivot out of it's dovetail socket in the scope base. Not a field-expedient maneuver.
Now, looking at the
other scoped rifle in the movie...
The U.S. Marine Corps used the National Match 1903A1 Springfield as their sniper rifle, with a 7.8x Unertl scope mounted to the receiver and barrel. They called it the M1941 Marine Corps Sniper Rifle when configured as such. Imagine this 1903A1 rifle:
wearing a scope like this one:
Which is a totally different arrangement than the Army's 1903A4. And even more awkward to remove and replace in the field. Here's a black-and-white pic of the USMC M1941 Sniper Rifle:
So, yeah, there was a good bit of artistic license taken in the production of
Saving Private Ryan.