Thought I'd follow up. Astoninglishly, I wasn't the first to think of the ring magnet. That's the Grainger magnet on the left, glued to a screwdriver bit. It does have the strongest magnetic pull and has the smallest diameter, so that's a win. The two in the middle are random examples from Amazon. They're both better than magnetic bits alone, but the red one is too big for tight spaces. The tool on the right is the Greenlee, and positively locks the screw in place for installation and removal. When you really don't want to drop that little screw, use the Greenlee. Comes in #1 and #2 sizes.
I have that Greenlee tool. It works, within limitations. If the head of the screw is too big, it doesn't work. You cannot tighten the screw completely, you have to start it and use a regular screwdriver to finish the job. The head is too big for tight spaces.
I'd rather use a longer 1/4" hex bit with a ring magnet
