My Idea is first you upgrade old version to new version.
1. Then you can take upgrade database backup.
2. And delete all upgrade files in ftp.
3. And upload your latest version directly (Which version you upgrade that version only)
4. Install and change the upgrade database.
5. If not working you need to change something.
Now is good. Because some magento project we made lot of files changes. So, we upgrade that is not working well. So, we need do this type. 🙂