Share quality timeless content
For this reason, it's often worth tweeting old blog posts as well. Of course you shouldn't do this merely for the sake of it. You should choose only good, timeless content to promote. If there are some old posts that don't quite make the grade in this regard, why not rewrite them? It will be good for the overall quality of your blog apart from anything else.
Tweet them sparingly
Needles to say, you wouldn't want to overdo this by tweeting the same posts time and time again. It could become annoying to your followers and you might end up losing some of them. That said, it's worth tweeting the really good ones at least a few times over several months to get the greatest benefit.
As is so often the case with Twitter, timing is relevant here. If you tweet the old post at different times of the day, different followers will be logged in and see it in their streams. If you have a lot of followers this could add up to a good number of new visitors after a while.
And as long as you keep growing your following, a few months down the track you'll have many more of them who are sure to have not seen these posts. So you can do the whole thing again.
Manual or automatic?
I know that you can have software to do this all for you. But frankly I think it's better if you just tweet your old blog posts manually. You can vary the wording of the tweets -- as well as the hashtags -- to get the best results. And if you are doing it by hand, it's very unlikely you'll do it enough to reach a point of overkill.
This is certainly a worthwhile tactic if you have accumulated a lot of content over the years that is now buried deep in your archives, and therefore tends to get little or no search engine traffic.