Besides editing with a critical eye, what can I do to practice getting the compression of expression that makes poetry, well, poetry? Since the poem is a narrative I'm focused slightly less on multi-layered imagery and slightly more on constructing a story with spare but meaningful details.
I'm not a poet by any stretch of the imagination, but for what it's worth ...
Perhaps what you need to do is impose more limits on your poetry to force clearer thinking.
For example, I find the limiting syllabic structures of haiku to be paradoxically freeing by forcing me to choose words very carefully to express what I have to say. As a starting point, I would therefore suggest you place some arbitrary limits on the amounts of syllables you can use for each line in your poem.
Another suggestion I have for your editing process is to use a methodical, almost repetitive process.
First, shift to paper-based writing for your poetry (if you haven't already). I find computers tend to be ill-suited for poetry, unless you're tracking changes. Write each successive draft on separate sheets of paper. For each draft, write out what you wrote before, but identify those lines that strike you as too wordy or incorrect, and focus on altering them specifically. Cross those lines out, write what you think of, cross out words etc. and fix them. It'll look like a mess, but you'll clean it up afterwards when you write it out again on a new sheet of paper. Also, don't worry if you alter the rhyme with your changes, because on your subsequent drafts you can focus on making a different rhyme.
There are good reasons for this approach, the most important being that you keep a solid history of what you've written, so if you go in a direction that doesn't pan out, you can go back to a previous edit. The second reason is because the act of writing it out time after time helps you understand the rhythm and nature of what you're writing, and it becomes much easier to spot where things fall over, e.g. too verbose, lack of rhythm and so on.