I run a web hosting business and have lots of experience with mail deliverability issues.
I can make sure your IPs are properly configured in DNS and add DKIM (if your setup supports) to your mail.
Once that's done, as long as your content isn't spammy-looking, mail will be delivered to inboxes.
Once the records are fixed and DNS has propagated, I'll test delivery to Outlook, Gmail and my own server.