Hi,
Templates do not fall under migration process. Teline III to Teline IV is a template change plus they uses different modules for the same functions.
If i was you, i would have done the following :
1. Take complete backup of current site.
2. Make a clone of the current site into a sub folder of the site.
3. Compare the download folder of teline III and Teline IV for the extensions and their versions.
4. Install Teline IV and t3 plugin to start with.
5. Uninstall and install the latest modules and configure them by referring to the userguide or best to install Teline IV quickstart in localhost to refer to the settings.
6. JA Zin plugin is no longer used in teline IV, so you will have to make the menu items yourselves.
7. Once done and settled swap the folders to make your new site live.
Hope it gives you idea on how to get it done.
Regards
Arvind