Send the e-mails from different addresses to different batches of your users.
LinkedIn emails get in because they have properly configured e-mail via SPF and DKIM, they are double opt-in, and they somehow work with google and microsoft. (Without double opt-in you get marked as spam more often. – I have no idea whether you already do this.)
Having malformed message-Ids where the domain in the message-ID does not match the sending domain.