What may be confusing you is that "dem Orkan" is used as a reference to a concrete point in time here.
aussieht wie direkt nach dem Orkan
is basically saying
it looked like [everything looked] just after hurricane [had passed]
I added the notes to make it easier to understand.
The sentence, however, is perfectly fine as "wie" can introduce a component part of a sentence and "direkt nach dem Orkan" is a "Zeitergänzung" (temporal component part).
Let's look at some more examples. Every temporal adverb is a valid Zeitergänzung, e.g. "gestern", "vorgestern", as is, of course, "vor zwei Wochen".
Der Wald sieht aus wie gestern / vorgestern / vor zwei Wochen
Now this doesn't sound odd nor is it hard to understand, is it?
About "ein Stück Wald". That's an indication of quantity, like saying "ein Glas Wasser", "ein Laib Brot". If you said
Ich hätte gern ein Glas des Wassers / ein Laib des Brots
you're saying that you'd like a glass of that (a particular, already previously mentioned or otherwise known to the speaker) water or a loaf of that bread. Apparently the author of the book you're quoting from is talking about some (any) piece of woodland or, if he mentioned the wood previously, he doesn’t want to stress that it’s a part of that wood.