Some ideas:
1) Since you're going for 3 sets, you may want to experiment with true cluster training with increasing rest periods. Basically, during 10s, you switch to cluster minisets of 3-5 reps, starting out with very short rest periods (15-30 seconds) and shoot for 30 reps. During 5s, with cluster sets of 1-3 reps, you shoot for 15+ reps starting with 5-15 second rest periods. Increase rest periods significantly as fatigue builds. This is roughly based on Vince Gironda's rep matrix ideas, albeit it's a bit lower in the overall volume, and it's not that dissimilar to the 20-rep programs you may have seen in the past.
2) Add in the burn set during 5s.
3) During post-5s, work with strong range partials during chins and rows and keep increasing weight.
4) During post-5s, see if you can bump up the load one or two more times for dips and military press.
5) If still unsatifsfied with your arms, try adding in loaded stretches that targets the arms.
cheers,
Jules