I have replaced the supplied EMG pickups with another brand, and there is a hum issue. Can it be solved?
EMG recommends to not include a string ground with their pickups, so when changing to another brand, it has to be added.
You can follow the instructions here: http://guitarworks.thestrandbergs.com/2009/10/24/ground-zero/ to create a string ground for the G string (historically, the high E string was used, but the alignment with the spring will generally be better on the G-string). Instead of drilling straight from the high E bridge to the control cavity, drill from the G-string bridge to the bridge pickup cavity, and route the cable that way.
The zero fret will ground the remaining strings. Depending on how your intonation is set up, you can usually nowadays get away with not grinding off any of the anodization, but simply wrap the ground wire around the front bridge screw of the G-string, since the saddle spring will sit right on top of it.