I have just restarted training MMA, about 3-4 times per week. This is normally a couple of sessions of gi-based BJJ, and a couple of no gi Vale Tudo or sub wrestling sessions. Nowt too serious, just a way to spend my time productively and make me feel good. As such my supplementary training may not be the ultimate way to improve my game, (I have other goals too) but it certainly doesn’t harm it;
at the moment I am training in the one lift a day style espoused by Dan John, who incidently should be paying me commission for the amount of posts I’m putting on the internet raving about his work.
I stick to compound movements - oly hybrids, powerlift hybrids etc, and train either heavy, (not necessarily max effort) or for speed - ie dynamic effort style. I finish with some anaerobic conditioning. At this stage of the workout, I may include some sport specific drills - eg pounding a prone heavy bag etc.
If you wanted to be really anal about your training, you could try this method for a few weeks:
perform a whole body lift hard and heavy
immediately perform a sport specific exercise as explosively as possible - obviously you may need a partner here for judo, although I guess you could simply throw a heavy bag or something - don’t think this training style would be applicable to newaza so much though.
This idea was used by Russian sprint cyclists, and I used it to increase punching power last year. While I am not a scientific type, and so cannot back up my experiment with solid evidence, I felt it worked a treat.