First of all, congratulations on creating a course that your students love! I am sure it is really rewarding for you to see that 31% completion rate.
Here's my take on this, as someone who has successfully sold online courses for the past 7+ years...
First, I wish I knew more about your goals. You mentioned that your course is free, so it makes me wonder whether you are doing this for charity, or if you are using this free course as a tool to nurture leads for another product or service?
In any case, there are tons of ways to encourage your students to become active promoters. Here are a few ideas:
1. Ask for students to review your course upon completion (or even upon completion of each module).
Not only will you be able to put these testimonials up on your website for social proof, but your students will start to think of themselves as ambassadors of your program, and will be far more likely to recommend it to a friend.
2. Offer rewards for referrals.
There are tools that will allow you to track referrals and offer rewards to people for successfully referring a certain number of students. Think about what bonuses you could offer, such as coaching or an additional lesson, and use these as incentives for students to share your course through their unique referral URL. Here's an example of a similar program: https://tim.blog/2014/07/21/harrys-prelaunchr-email/
3. Use your free eBook as a teaser.
Don't give away the farm in your eBook– keep it very brief, actionable, and quick to digest. A checklist is best for this. Then at the end of your eBook, include a call to action to enroll in the course.
4. Use competitions to give away bonus content.
If your course was paid, I would say to hold a contest where the winner earns a free scholarship. Since your course is already free, however, you could offer a bonus as the prize. Again, this could be coaching, access to a private group, an additional lesson... anything that will enhance the course.
Those are just a few ideas to get you started... the possibilities are endless!
I hope that helped. If you want to chat further, feel free to give me a call. Good luck!!