Chin ups between sets of presses is a great approach. Lots of great lifters liked "paired sets." Arnold, Paul Carter, Paul Anderson, Alpha. That's just like, the A's.
I would do non-killer sets of chins, worked into the WS4SB upper days.
Do like 2 reps with a little weight added between pressing sets on max effort days. You can do this right thru the Secondary exercise too. That could be like 8-12 sets right there. It wouldn't even be hard. Over a few weeks, you increased to 3, then 4 chins.
It won't take you very long to hit The 30 reps weighted in 10 minutes, in addition to your pressing.
Do unweighted sets of 4-6 chins between presses on rep days. Again, non-killer sets. You can keep it up during the Secondary exercise again. Thats 6-8 sets. Let the volume add up. Over a few weeks, build up to like 5-7 per set.