Well first, be sure with your new host that you get regular backups going 😉
Towards the bottom of my explanation, I’ll clarify how you can potentially retain/enable your other clients to use the old hostname so you’re configuration impact is minimal.
Next, be aware for your new host, you can make use of “Let’s Encrypt” to obtain a free certificate for your host. I would recommend naming the host something unique from the original since you already have some other certificate issued for it.
With your new host setup, (make sure A/AAAA and rDNS are also setup for the new host), you can then add/replace the MX record for your domain to point to this host. Take out the old host from the MX record list.
Now you have at least a routable method for email delivery for your domain.
Next, delete the A/AAAA DNS record for mailinabox.email (your old host), so you can recreate the record as a CNAME alias.
Finally, create a CNAME alias record that maps the mailinabox.email -> mail.onze.host (maps the old hostname to the new hostname.) Now your existing email clients that use the original hostname should continue to work because they will find the new hostname via the old one.
If you have a DKIM/SPF/DMARC records in DNS, be sure to validate those as well. Setting up a new email server generally results in a new DKIM private key, so the public key in DNS needs to be updated.
Hope this helps!