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In theory, when a question is voted closed, then the OP edits the question to be site-appropriate, the question will be added to a review queue.
However, in practice, the question, now made good, may possibly never make it back to the land of living questions, due to the original swarm of downvotes.
If a user cannot reclaim the now-good question into an upvote situation, what should they do to revive interest in the question?
One problem with considering it "ok" to re-post instead of going through the reopen process, is that most people that ask off-topic questions do so because they don't understand the posting rules, or the scope, of the site they're asking on.
The Stack Exchange system (not its moderators, not its users: the system itself) will automatically block a user from asking new questions, if they can't be bothered to learn the rules and play by them. The details are unknown, but shortly put, if you have closed, downvoted questions, deleting them won't fix your "standing" in the system - deleted posts still count towards an automatic ban.
The idea is that users are encouraged to edit their bad posts into shape, rather than deleting and re-posting. You see some people don't care about the rules, and will re-post a crap question all day if they think that's how they're going to get an answer. The system protects itself specifically against that, and vigorously.
If your question was put on hold, it's very likely that at least one user commented on it (often with a canned comment, if the question displays a common misunderstanding of how Stack Exchange Q&A sites work). When you edit your post, @ping that user to let them know you've read the link(s) they provided to /help/how-to-ask or /help/on-topic, and have edited your post accordingly.
If you did get your question back on track with that edit, a close vote can be retracted, a downvote can be reversed, and an obsolete comment thread can be wiped out by a moderator.
If an edit is suggested to your question while it's on hold, and that edit does not make your question on-topic, reject that edit: only the first edit made to a question after it's put "on hold" will make the post enter the "reopen" review queue - if the edit isn't making the post on-topic, the reviewers will vote to keep it closed instead of reopening it, and when you do edit it into shape, your question will not make it to the reopen queue, and then yes, it can become pretty hard to get your question reopened.
Don't delete your downvoted posts, edit them into shape instead. Otherwise you risk falling under some threshold and blocking yourself from asking further questions. Not even moderators can lift an automatic question ban: you're in charge of your own standing.
As for the "initial swarm of downvotes" - deleting the post only makes it worse. If you have enough reputation score, you can put up a bounty to draw attention: if it really is a good question now, that doesn't deserve such a negative score, the bounty period should help turning it around.
In any case, the single best thing to do is: don't do it again. Read and understand the /help/how-to-ask and /help/on-topic links on the site(s) you're asking on, read some upvoted questions on that site, see what they all have in common, and make that an ingredient in your own future posts.
No, it's not appropriate to just re-ask your question when it's closed. The reopen process is there for a reason (to ensure that questions really have been fixed before answers are posted to them) and trying to subvert that process by posting duplicate questions is not appropriate.
