I then uploaded to the new server in a test subdirectory and unpacked the compressed file via the control panel file manager. The upload took the usual 20 minutes. (A question: is secure technology available to do a straight old-server to new-server file transfer without using one's local desktop as an intermediate stopping point?)
I then pointed my browser to the installation subdirectory created during the unpack process. This runs the install program. I had a few password related problems.
Then I remembered that the new server used a different prefix ... old MySQL db was xxxx_site1 , new MySQL db was yyyy_site1. Old MySQL user was xxxx_admin, New MySQL user was yyyy_admin.
So I got around this by
1. Manually creating the new db 'site1' on the new site using the cpanel. This, of course, became db 'yyyy_site1' automatically on the new server.
2. Manually creating the user 'admin' on the new site, which got translated to 'yyyy_admin'. I gave it a password and wrote it down.
3. Manually adding the new user 'admin' to the new db 'site1', with full permissions, naturally.
Then I reran the install program with the MySQL user named 'admin' and the password I had written down.
I spent another few minutes checking out each page, and everything looked good at first glance. Two days later things still look fine.
Perhaps I took the long way, but, even with the few extra steps, it was very easy and fairly quick. My sincere thanks to the author and to those whose testing has refined the component.