score:6
No, there is no waiting period before you can apply for a visa again. Your new visa application will be evaluated on its merits. Applying for a visa with a longer validity period will not matter much.
However, in your new application you should absolutely explain and document why your earlier visit ended up longer than you stated in the previous application.
We have several examples on this site on travelers to the UK who stayed longer than they said they would, even if their actual stay was within what they were actually granted, and then had a subsequent visa application rejected because the difference hurt their credibility. (In many of these cases the difference between planned and actual stay was more than in your case, but still almost doubling the length of stay approaches the point where one can reasonably expect eyebrows to be raised).
What you need to do to avoid getting into that kind of trouble is to proactively explain in your new application how and why your plans changed after your previous visa was issued. If your explanation looks reasonable and you have documentation that is consistent with it, that will save your credibility. Do not wait until you're asked for an explanation, because you won't be asked -- you just get a refusal based on "You have not explained why such-and-such".