How small and precise do you need?
I've seen Rapidograph pens used as clock and watch oilers. You could oil the tip of a cut hair with one of those, and not get anything on the sides.
I have one of those oilers from General (they're carried at most hardware stores nowadays), and a few with no name that are 30+ years old. The design hasn't changed, and looking online, I see them rebranded by dozens of companies. The Brownells version is clearly made by the same people. It is a very good oiler, but makes a largeish drop.
I like the fact that that Beta one is clear, so you can see how much lubricant is left, but other than that, it is the same design as the General, and has the same issues.
I have some old plastic clock oilers which have a similar body design, but instead of a button on the top, the tip pushes in, like a paint pen. Those will leave a smaller drop, but I can't find any brand name on them, and have no idea if they're still made.
Edit: Found it, google this:
Bergeon oiler #4722