If you invited your participants via Doodle, go ahead and click the 'invite' button at the top of the page to invite more people. Click 'email' and then add the new email addresses. Only the new participants will receive an invitation. No email will be sent to the previous invited participants.
If you invited your participants via the link, you can simply send out the link again to any additional participants. You can click 'invite' at the top of the page and then 'link.' You can copy the link and send it out as you like.
Make sure to read here as well: How to invite participants to my poll?