Just for fun, I would clip a wire directly to the ground lug on the output jack, then touch it to various places that should be grounded to see if the hum goes away. For example, touch the pot backs, then the strings themselves. If that calms things down, then you'll know you've got a missing, open or high-resistance connection somewhere. Use an alligator lead, such as one of these...
You can
buy a set of 10 here for about $4, and they're handy to have around.
Then, I'd question what you mean by noisy. Those look like single coil pickups to me. Even when wired to behave like humbuckers, they're not as quiet as a real one. When used as single coils, they're noisy as Wayne County lockup. It's the nature of the beast. How could the guitar have been quiet before? If it was,
that's what I'd be suspicious of.