tl;dr: way more Volume, weighted chins, prehab upper back stuff, abs abs abs... or alt: fuck all that and ask ed.
T-Nation DemiGod Alpha's method on improving pullup performance
Former Navy SEAL Stew Smith's method for doubling your pullup performance
Since your reps are lower, it's usually good to start off improving yourself structurally. Especially from your log it seems like you have an issue keeping your upper back tight. So I would also do some research for:
- Face Pulls, face pulls, more face pulls
- Weighted Negatives
- Iso holds (at various positions/weak pull points, usually, dead hang, very top or midway)
- Band assisted variations
- ytwi drill (rear delt drill)
- side bends, holding planks and anti flexion ab exercises for time (stronger abs ∝ pullup strength)
- tennis ball rolling trigger points in rear delts
- Increasing your overall daily volume, 3x/wk, then 4x/wk, then daily, but just small additions at a time, give your body a chance to acclimate first then add more volume.
If I were in your position... and if your strength allows for it... I would utilize stew smith's method but instead of the pyramid day I would throw in weighted chins. I like 10 sets of 1-3 straight across. Think like wendler, it doesn't have to be your damn max, but 10-30total reps with 10lbs is going to do more for your strenth-wise than the same thing with just bodyweight (ie, a lighter weight). I'm always going to be in favor or getting stronger. But if you can't do weighted chins yet then weighted negatives sub VERY well in this regard. They are a bitch to setup sometimes though, but be imaginative. Also I would make sure you're doing your ab work. A lot of people have issues with swaying or laterally or front to back and usually related to weak abs and just not staying tight. That will NOT slide at west point lol.
BUT WHAT I WOULD REALLLLLLLLLLLLLLLLY DO IN YOUR POSITION...is ask your Olympian god-like coach for his advice and solution to this issue since you are currently under his masterful tutelage. What I would not do is let the advice of some random ass internet stranger (such as myself) fuck up Coach Coan's awesome programming. I'm 1/2 joking but 100% serious... you should ask Ed. Good luck!