OK, I think I have this set up as good as I am going to get it now.
I first fixed the hanger alignment using the second wheel trick:
It wasn't horribly off, but a little.
Unfortunately, shifting didn't improve. How can that be? I think the explanation is that I have my chain shorter than it's supposed to be, in order to not lose too many gears on the small chainring. The derailleur cage is designed in such a way that as it slackens (on smaller cogs), the upper pulley comes closer to the cassette, like an automated B-screw adjustment. Due to the short chain, my derailleur doesn't get enough slack.
So I adjusted the B-screw more, bringing the pulley closer to the cassette until it shifted properly. The cable is as tight as it can be without catching on the large cog when in second gear. Now shifting seems to work properly from the smallest cog to the lower gear, and also back the other way. And I can still use the largest cog, even if the chain is sandwiched between the cog and the pulley.
The MTB cage I bought seems to be identical with regard to the placement of the upper pulley, so I don't think it's going to improve this. But I'll give it a try next week. If nothing else, it will allow me to use a slightly longer chain, which should improve things.
Also, it seems like switching to the 11-42 might be a stretch. However, it should work if I also switch to a standard 53-39 crank. Why would I want to do that? Because I do have a 10-42 cassette waiting to be put on, which would give me a higher top gear. If there was such a thing as an 11-speed 10-36 cassette, it would be great, but there isn't. There is, however, a 12-speed 10-36. But 12-speed has even more finicky gearing than 11-speed, so I'm not going that route.
Another potentially crucial detail here is that my 11-36 is not Shimano, but SunRace. The 11-42, on the other hand, is Shimano.
