OK, got it working smoothly. The cassette spacer was the culprit!
I started with re-tensioning it, step-by-step as you suggested. It was a reasonable place to start given the limited info I provided. Once I had the cable re-tensioned, however, I was back where I started. That is, when the cable tension was correct for the large/inner cog, the derailleur wouldn't quite shift all the way back down to the small/outer cog. Vice-versa if I adjusted for the inner cog - it wouldn't quite stay when shifted all the way to the large/inner cog.
Next, I played with the tension screw, just in case. It seemed to get slightly better as I got the pulley closer to the cassette, but it clearly didn't solve the problem.
Then, I thought through the problem a little further and wondered if there was something funky with my cassette. My first hypothesis was maybe the small/outer cog was not seated properly on the splines. I removed the wheel, but the cassette looked fine, cogs evenly spaced.
Finally, I removed the casette with my cassette tool and decided to try removing the spacer. This is a Shimano Powertap hub with 10-speed Ultegra cassette, fairly new. I put the cassette back on without the spacer and checked to ensure there was no play. Re-installed the wheel, re-adjusted the tension with the barrel, and FINALLY got it working again. The confirmation to me was that I was able to re-adjust the H-set screw, and it is no longer sticking way out.
That was the solution - removing the spacer.