I'm having a *lot* of trouble seeing through the rear aperture on my Springfield Standard M1A.  In a average lit indoor range, it sometimes feels like a guessing game as to where I'm shooting.  It's the factory rear sight, non-hooded.

I'm probably not the best with calipers, and this is a cheap digital one off Amazon, but I'm measuring .057" measured from the flat side.

Is ~.057" the right size for a USGI M14?  I looked on Fulton Armory and they're selling what appears to be a standard size rear sight, which they say to be 0.070".

 Match apertures are .052" and .0595".  Garand apertures size is .069 - .074", and may also be for a standard M14.

Is there a light shining down on the ap from behind or so?  

To me that's when a small ap is a bad deal.  When the sun is to your back.  If you wear a hat you can overcome that though.  With a bill to shade the ap.  But light shining down on the ap lights it up like a christmas tree and then you can't see the front sight.

But yeah, you might have a NM sight on there and it's smaller than a GI.  I put a GI on my M1A at present.  But the NM will be more accurate for just bench shooting.  Of course.  I'm not really sure how much though.  The NM sights are typically tighter in the rear sight channel too.  Making them a better deal as you dial in more elevation.

Is it clean and dull?  The only time I have had problems seeing through those sights is when they get shiny or something in the hole.  It looks like a clean hole when you look at it, but if there is oil, dirt, or grease in it, depending on how it got in there, it could be clinging to the inner diameter of the hole making it smaller and gumming up the works when you try to look through it.  I had one that I could not see through very well even though the hole looked clear.  I finally stuck a q-tip in there and gave it a good cleaning.  It was good to go after that.

I have NOS unissued apertures if you need a new one. PM me

The NM ramp w/hooded aperture are made oversize. They are not intended to be inserted and used. They should be stoned to fit the particular base that they will be fitted into.

This is a good idea. Even the tiniest obstruction you may not be able to see will be magnified when you look through the sight.

You want the inexpensive and quick fix?

If you have a good set of drill bits and a drill.  Drill out the sight to the size that fits your eye's ability.    A little touch up blue on the inside edges to defeat the clean metal shine or even a dab of dull black paint.  I did that with my RR NM 0.040 diameter rear.  On any day that wasn't full sun it looked like there was fuzz/cobwebs in the aperture.  I had to go bigger about 0.062".  The older you get the more you might have the issue with the tiny apertures.

Is it the best option?  That's up to you.  A stock USGI rear sight is an easy install and reversible.