Aside from having a catchy subscription process on your website, you can use calls to action on your social media platforms to attract new readers. While doing a little research recently on Twitter, here are some examples I found - bad examples first.
I have no idea why I would want this newsletter - I'm not even remotely curious to know what those 'important updates' are.
Aside from having no interest in 'the latest' from a stranger, these poor folks don't even have a sign up form. I wonder how many new subscribers they get.
I could show you hundreds of examples like this - or you could go look yourself. Doing this is a total waste of your time.
Now for the good examples:
Authors generally do a pretty good job of creating interest with their calls to action.
This one is short and sweet. You know it's a newsletter and you know what it's about.
This seems like a pretty good reason to sign up. And you can see it got retweeted. Too bad they spelled Christmas wrong.
screen shot at top from shinesty.com