Okay, the setup is now the following:
Host old host: domain.com
IP address old host: x.y.z.w
New host: new.domain.com
Forwarding old host -> new host is done with exim4 with a manualroute and remote_smtp transport.
domain.com has been added to Mailcow UI -> Configuration -> Forwarding Hosts
If I send an email from a public email provider (gmx.net) to my private mail address at firstname.lastname@example.org I get R_SPF_FAIL for mails when receiving them on new.domain.com and this message in the headers:
spf=fail (new.domain.com: domain of email@example.com does not designate x.y.z.w as permitted sender) firstname.lastname@example.org