I discovered my server was on a blacklist from Microsoft, probably because of the provider. I couldn’t get it whitelisted as fast as I would, so I configured an SMTP relay for these specific domains. The error message from Exim:

F=<***@domain.com> R=lookuphost T=remote_smtp H=hotmail-com.olc.protection.outlook.com [104.47.14.33] X=TLS1.2:ECDHE-RSA-AES256-GCM-SHA384:256 CV=yes: SMTP error from remote mail server after pipelined MAIL FROM:<***@domain.com> SIZE=165698: 550 5.7.1 Unfortunately, messages from [XX.XX.XX.XXX] weren't sent. Please contact your Internet service provider since part of their network is on our block list (S3140). You can also refer your provider to http://mail.live.com/mail/troubleshooting.aspx#errors. [VI1EUR04FT004.eop-eur04.prod.protection.outlook.com]

To relay the mail to an Exim SMTP server I had setup for this, allowing only access to the SMTP ports (25,465,587) by IP whitelisting, I configured the following config for Exim:

Create a new file called /etc/exim.routers.pre.conf

And configure it as following:

NOTE: If you don’t want to forward specific domains to the relay, but all, then replace domains = xxx with:

domains = ! +local_domains

This will forward all domains to your SMTP relay.

smart_route_forward:
    driver = manualroute
    domains = hotmail.com:live.nl:live.com:outlook.com:hotmail.nl
    ignore_target_hosts = 127.0.0.0/8
    condition = ${if !eq{$original_domain}{$domain}}
    condition = ${if !eq{$original_domain}{}}
    condition = "${perl{check_limits}}"
    
    transport = remote_smtp_forward_transport

    route_list = * smtprelay.domain.com::587
    no_more

smart_route:
    driver = manualroute
    domains = hotmail.com:live.nl:live.com:outlook.com:hotmail.nl
    ignore_target_hosts = 127.0.0.0/8
    condition = "${perl{check_limits}}"

    transport = remote_smtp

    route_list = * smtprelay.domain.com::587
    no_more

After this, restart the Exim service:

service exim restart