First of all, analyse your best-performing tweets (most liked, most retweets, comments etc)
Then make a list of order (the order should be highly engaged tweets to least) and group them into topics or cluster the same tweets.
Once you have the list, it's time for execution; find out the most active days when your followers are online and at least find out when they engage with your time.
Now retweet as a mixture of content (someday retweet most performing tweets of yours and other days least performing)
Remember; day & time are absolutely important. For every account, it works differently!
Use your old tweets twice a week, not more than that! No one wants to feel like there feed is a spam feed!
Keep your eye on trends, and topics and use Twitter threads. And engage with your followers, with their tweets. Make them feel important! Connect with them on a personal level.
@EJudson can you let us know what you mean by "new people" -- have you added a lot of new followers recently and you want to retweet something from before all of them started following you, for example?
For context, I'm running the account for an alumni relations department at a university. We've added a few followers recently, but I was going to retweet a tweet that contains a link to register for a webinar. I'm hoping to get more people to sign up.
I'd say the best way to "retweet" a tweet is by via quote tweeting where you can add more value to the tweet, plus it would give double the traction. Would be my best advice.