This is only an extension to a normal installation. So you should have your new Zimbra environment ready to move the data of your current Zimbra environment to it.
In the past we did an in-place upgrade of our Zimbra and the Ubuntu it was residing on. The last upgrade from 8.5.1 on Ubuntu 14.04 to 8.7.1 on Ubuntu 16.04 was such a drama that I started looking for an easier solution that would give me a fresh Zimbra with all my settings, accounts, mails, calendar items ... without too much fuss.
To prevent inconsistency please stop your old Zimbra and afterwards make a snapshot and/or backup before you continue.
su - zimbra -c "zmcontrol stop"
Also stop and disable fetchmail if you're using it
sed -i "s/START_DAEMON=yes/START_DAEMON=no/g" /etc/default/fetchmail
Next change the IP because the Zimbra will need to be started again to do the export
sed -i "s/old_ip/temporary_ip/g" /etc/hosts /etc/network/interfaces
Next on both Zimbras create a folder to store the migration data on
su - zimbra -c "mkdir /opt/zimbra/backup/exports /opt/zimbra/backup/zextras"
Next mount this folder as the user zimbra on the new Zimbra (don't set a password on the zimbra user, just create a key pair and share the SSH public key). This will save you time to move the data.
sshfs zimbra@temporary_ip:/opt/zimbra/backup/exports/ /opt/zimbra/backup/exports/
Next install the migration tool on the old Zimbra
cd /tmp && wget https://download.zextras.com/zextras_migration_tool-latest.tgz
tar xvzf zextras_migration_tool-*.tgz
Now open the Zimbra admin page (https://temporary_ip:7071) and start exporting the domains and user you want to keep. Through ZeXtras -> ZxMig -> Start Migration
Change the folder to the newly created /opt/zimbra/backup/exports
Select the domains you want to move
Under ZxNotifications check for any error during the export
When this finishes you can start importing on the new Zimbra. First install the Zextra suite (this is free for 30 days and we will remove it after we finish to get rid of warnings afterwards)
cd /tmp && wget http://download.zextras.com/zextras_suite-latest.tgz
tar xvzf zextras_suite-*.tgz
Open the Zimbra admin page (https://new_ip:7071) and open the ZeXtas -> ZxBackup -> select Import Backup
Change the path to /opt/zimbra/backup/exports
Select the domains you want to import
Select the accounts you want to import
And check if the restore started without warnings
After this finished you can clean up and give your new Zimbra the IP of the old one to start sending and receiving mails again.
If you had an SSL certificate installed now is the time to move that as well. Do this as user zimbra!
On the old zimbra
rsync -au /opt/zimbra/ssl/ zimbra@new_ip:/tmp/ssl/
On the new Zimbra
cp /tmp/ssl/zimbra/commercial/commercial.key /opt/zimbra/ssl/zimbra/commercial/
./zmcertmgr deploycrt comm /tmp/ssl/zimbra/commercial/commercial.crt /tmp/ssl/zimbra/commercial/commercial_ca.crt
In any case I would also enable HTTP -> HTTPS redirection
zmprov ms "FQDN" zimbraReverseProxyMailMode redirect
FIRST: shutdown the old Zimbra machine and make sure it won't boot by itself. Don't forget to copy any backup settings, fetchmail config, cron jobs, ...
On the new Zimbra
./install -u all
sed -i "s/new_ip/old_ip/g" /etc/hosts /etc/network/interfaces
Now your new Zimbra should be available with all the settings, accounts and mails just like it was when you stopped your old Zimbra.