If we use the assumption (assumption only, you'll need to test and tweak

) that you can convert at about 10% when you try and convert
targeted visitors on a free item, then you might need to put your offer to sign-up for whatever you're giving away free in front of at least 100% to 200% of the number of signups you need.
So if you show this offer to 5500 (at 10% conversion, you might get 550 people sign up for the offer, if you can only convert at 5%, then it will be half of 550).
Get in touch with email list owners in the same or very related niche and ask to do a JV with them perhaps. If they are willing, perhaps ask them what their past conversion rates have been, whenever they have mailed out an offer to them.
At the end of the day, if emailing to 5500 isn't enough but it works, you might want to email more prospects
Cheerio
What's the fastest way to get 550 people to create an account on a website?
Thanks for any input,
Tommac