A lot of puzzles end up getting quite a few partial answers, hints, etc. before the actual puzzle is solved. If this is unhelpful, it eventually gets down-voted. However, often this content is, in fact, helpful to solving the puzzling.
The problem with such answers is, once a question with an answer (with at least a few up-votes) no longer shows on the unanswered tab. On Puzzling SE, there are limited questions right now, so many people continue to try solving it; however, when there are many such questions (in the future), people will often forget about them. Furthermore, if a question has 2 (or more) parts, and only the first part is solved, some people forget that there is a second part left. The OP accepts the answer to the first half, and the rest of the question is forgotten.
The least we can do is to ensure that they remain in the unanswered tab, so someone browsing through certain tags in the unanswered section might notice it and put a bounty, or else post an answer himself, to get some attention on the question.
For this, we can introduce a check-box called 'partial answer' (somewhat like the 'community wiki' check-box) when posting an answer. A question should remain in the unanswered tab, until there is a non-'partial answer' with up-votes. The person posting the answer and moderators should have access to changing this, after the answer has been posted. I'm not sure if the person posting the question should be able to do so on an answer himself.
This would be suitable on quite a few sites, not just Puzzling SE, so I tried asking on Meta SE. However, I had made some bad posts earlier and have been banned from posting questions (even though people liked my last question, which was a feature-request that got denied).