It's an NOS Archer 15-1273A with a 2-piece sheet steel housing (the base that the PCB and transformer mounts to and a cover that attaches to the base with 4 screws).
I removed the cover to see if I could find the source of the humming, but when I plugged it in without the cover, there was no hum whatsoever. I put the cover back on and the hum returned. I repeated that enough times to confirm that it only hums with the cover on (even with the 4 screws removed).
Why would the cover cause it to hum? It doesn't contact anything except for the sheet steel base.
Is there any way to stop it from humming short of leaving the cover off?
Transformers operating on 50 or 60 Hz mains tend to hum and the vibrations get conducted into the case which can resonate and make them louder. You might be able to reduce by putting rubber between the transformer and case and mounting bolts.
The hum may be annoying but it does not mean that anything is faulty.
if you weld together a mild steel flatbar in the shape of a rectangle and mount the transformer in there it might prevent the field from rattling the chassis (magnetic shunt).
I noticed a big transformer when tested on top of a piece of sheet metal made alot of noise. when it was on a nickel plated steel base it made less noise
Make sure the two screws holding the transformer down are tight.
The cover seems to be the "speaker" here, so maybe a piece of rubber or something glued to the top of the transformer, so it pushes up on the cover some to discourage it from vibrating?
I don't think the cover is significantly acting as a speaker cone, because if it were, gripping it tightly with my hands should dampen its vibrations and reduce or eliminate the humming, but doing that doesn't affect the sound of the humming at all.
I tried putting a sheet of thin plastic folded to match the shape of the cover (a transparency sheet for inkjet printers) between the cover and the base as an insulator/separator, but it made no difference at all. I also listened with just the clear plastic being used as a makeshift cover while holding it tight against the bottom sides of the base, and even that produced hum, though quieter than the hum you get with the steel cover.
I think that just enclosing it with anything is creating a resonance chamber or something to that effect.
I also tried putting rubber washers under the transformer's feet, as well as under the heads of the two mounting screws (4 rubber washers in total), which completely isolated the transformer from direct metal-to-metal contact with the housing, but that didn't make any difference at all.
Putting rubber on top of the transformer (as suggested by BrokenYugo) that's thick enough to press against the cover does quiet the hum by a noticeable amount (no cover at all is still quieter though). I'm guessing that the metal housing of the transformer is the primary "speaker cone" here, but it doesn't make any noticeable sound unless it's in a small, enclosed space. Rubber being pressed against the top of the transformer would dampen its vibrations some.