When you go to add an addon domain in cPanel, a script is run to check the WhoIs database and see if the domain being added is pointing to the correct server. If the propagation process hasn't completed yet, you will not be able to add the domain in cPanel yet. You can shorten this problem by doing the following:

Let's assume the addon domain is cat.com and the primary domain is dog.com

1. Log into the cPanel for dog.com

2. Click File Manager

3. Click the yellow folder icon next to public_html

4. Click Add Folder. Name the folder cat

5. Upload all files for cat.com to the new subfolder cat

6. Open a browser and go to www.dog.com/cat. Verify the site is working the way it should be

7. Once you are satisfied the site is functioning correctly on the new server you can now go to the registrar and switch the name servers

8. Go into cPanel and try adding the addon domain. If it will not add the addon domain, propagation is still working and you just need to keep trying. Propagation times have dramatically reduced in the past year, so give it an hour and try again. Keep trying every hour until is successfully adds the addon domain

NOTE: This will not eliminate the propagation process, but as soon as it completes the site will be live. Nothing can be done to eliminate downtime due to propagation, but following these steps will ensure the shortest downtime possible.