What are some strategies for promoting the site outside of the Stack Exchange network?
Some ideas (to be expanded on later) ...
Post ads on other forums.
Someone already did this for Lit while it was still in Area 51. We can design a new description of the site, preferably something short and snappy which sounds attractive but still makes clear the nature of the site (Q&A not discussion, etc.), and post it on various other literary forums.
(Of course, make sure to check the promotion rules for any site you aim to advertise on, so you don't get yourself banned for spamming. That goes without saying.)
Tell real-life friends about it.Know someone who's a book critic, or a literature student, or a librarian, or just really keen on reading? Tell them about this awesome website you go to and how we're trying to build up a repository of knowledge on all things literary!
I'm not a social media person myself, but from what little I know about it, it seems to be a potential way to advertise oneself or one's project and even possibly get in touch with famous people in the field. In the case of Lit.SE, for instance, you could tweet about particularly interesting questions, or contact authors/critics/etc. asking them to sign up and provide answers.
Edit: some of our honourable community members have valiantly stepped up to the task and made us a Twitter account: Literature now has a community-maintained Twitter account!