> No such thing as a free lunch.
Correct. And Shure already ate all the free breadsticks.
> Lower self-noise?
The SM-57 is transformed to about 300Ω, on the high side of mike impedances, to get maximum output (and S/N) with typical mike preamps. You are not going to find a significantly better ratio.
Are you SURE the noise is the mike, not the preamp or the room? Self-noise of a '57 is around 20dB SPL, and many rooms are noisier than that.
> A reduction in hi-gain hum?
50/60-Hz hum is a different problem. Get away from power lines and AC cords.
Pitchless "hum" is the room ventilation or distant traffic noise coming through the wall. Or maybe flicker noise in the mike preamp, though this is usually overwhelmed by room rumble.
Recall that most speech/singing has a specific lowest tone. Most female singers, you can low-cut at 220Hz and lose none of the singing (I have met exceptions). Male singers can be strong to 80Hz, but a steep 70Hz low-cut can really clean up the sound if room-rumble is a problem. Perhaps the worst problem is small-chest males who want to sing chest-tones, but can't power-boom. A single-D cardioid or bi-di mike is their friend, boosting nearby low tones without boosting distant room-rumble. Even so some electronic bass boost may be needed to give the ballsy sound they think they should have, and then you have to have a low-rumble room (or a booming bass track to smother room rumble in the vocal track).
> Flatter response?
Why use a '57? It is very good for many situations. If it isn't, get something else.
There are a host of '57/'58 imitators, with similar tonal sound, and several use super-magnets for improved output. I mostly use the AudioTechnica imitation '57/58s: the cheap ones are frankly inferior to a '57 but good enough for live jazz. The good ones (from A-T and everybody else) are nice variations on the '57 and some are higher output.
IIRC, the capsule in the '57 used to be about 10Ω, so the tranny musta been √(10Ω:300Ω)= 1:5. Note that it isn't just the ratio: a 1:5 150Ω:4KΩ transformer will give terrible response and noise. 10Ω:300Ω is going to be a custom winding. And for all I know, Shure may have changed to a transformerless capsule in the 20 years since I last opened one.