Turns out it was trivially easy to make a soldering iron holder out of a couple pieces of scrap wood, some screws, and a "spring grip" -- one of those spring grabby things you can push a broom handle into to hold it in place against a wall. Tools: saw, screwdriver, drill.
Step 1: Cut the Wood
You do want to assemble the finished tool so it holds your soldering iron at a good working angle, but doesn't obstruct the cord of it. I found that 45 degrees was too much, and used an angle closer to 30 degrees. The base only needs to be big enough to screw the triangle to it, and have some stick out for the clamp to attach to.
Step 2: Assemble
Step 3: Use
Here's me soldering, with the iron held fast by my lovely new tool. It worked great!
As suggested in the comments by jiovine, it may be even better to use the opposite orientation, where the iron is pointed down and towards you. I tried this orientation in a dry run and I think I will probably use it that way instead of with the iron tip pointed up in future. (The hot bit is closer to Me that way, so a bit more care will be required.) It's a little tight because I didn't design my holder with this orientation in mind, so if I have to use it for larger pieces I'll probably have to modify the holder. But for small work it's just fine.