It's pretty unusual to have hum with your kind of setup...
On the Control side, the CODEC gain settings need to be appropriate to the mic you are using: higher gain for a dynamic mic, lower for an electret. And on the radio side the same is true, but usually it's more a problem of avoiding overload (excessive gain) rather than not having enough gain and ending up down in the hum/noise world. Also set the CODEC HPF to something like 163 or 204 Hz. Might be worth a few tests.
As Mitch said, solid connections everywhere are always important. A bad shield connection would certainly be bad news.
I assume you're using only the shortest possible audio cable connections, i.e., no extensions.
One other thing comes to mind and that is proximity of the RRC to something containing a large power transformer, which can magnetically couple hum into audio equipment.
73
Gary NA6O