Very sloppy shifting on smaller cogs (BSO) I have very few complaints with my BSO, but this is one of them.

My rear derailleur seems to hold the same 'up and down' tension across all the cogs. So when I'm on the first couple of cogs (32 and 26 teeth), the shifting is precise and efficient. But when I move out towards the end of the freewheel, I have to push the lever a long way to make the jump to the next cog, and then pull it back to keep it from making noise.

Basically, the top jockey wheel stays at the same height all the way across the freewheel, which means that when it's on the small cogs, it's hanging well below them, and has to move a long way to get the chain to hop to the next cog.

Is there a way to adjust this, or is it simply a consequence of having BSO-level components?
OK Based on comments - your BSO rear derailler is not following the contour of your cassette, and is ending up a long way from the smaller cog.

It should be like this stylised image:



Doubtless you have some wear on the cassette, and the RD position is limiting the number of teeth that are actively engaged in the chain.

I bet on the big cog you've got 5/8 of the teeth meshed with the chain (~20 teeth) and on the little 14 you might have as few as 3/8 (~5 teeth)  so this accelerates wear, and fewer teeth meshed means skipping errors earlier.

You could try a new cassette and chain, if the cassette appears worn.

The other option would be check the rear derailler and see if you can make it move more diagonally

Another possibility is that your B-tension screw is too loose, or isn't even there.