I would advise doing an upgrade before you move it. This way you can hit the ground running the minute you're fully loaded on the new server.
If your php files are heavily customized - even in a "custom" template, you still need to do a file comparison check between NEW 1.3.8 files, and their customized (1.3.7) counterparts in your site. There will probably be only a handful of changes to CORE CODE - not the customized bits - as the versions are close.
This COMPARISON should be done after FTP-ing all your customiszed files to your local machine, then using a file comparison tool like WINMERGE, see if there are any critical differences between the new 1.3.8 files, and your custom files.
Where there are core code differences, apply the NEW code to your customized files.
This can be laborious if a lot of customizations were done, but it is necessary.
You can then create a full install fileset (which can include your updated custom files) by putting these custom files into appropriate folders in an unzipped copy of 1.3.8a.
While you are about it, you may as well apply all the patches tha are relevant.
Then, you will have a full FILESET that is 1.3.8a, up-to-date with your own custom files AND the patches.
This whole lot stays on your local drive for now...
Then, on the day you want to move the site, put it into maintenance mode to prevent any further data being written to the database. Then do a full Upgrade from 1.3.7 to 1.3.8. (Don't worry about out-of-date php files on the SERVER... you won't be using the stuff on the server - - - you have a ready-to-go set on your local drive.
The purpose of this upgrade is chiefly to get the database properly patched.
The minute the database is patched (upgraded) back it up ready for the move.
I don't normally bother running the installation program on the new server. I just create an empty database and then import all the data from the one on the old server (which of course is now 1.3.8.)
Then I ftp the FILES that are on my hard drive to the new server.
I manually adjust the configure.php files and the dbase cache record and make sure all hard-coded paths are correct (if applicable).
Then the new site is basically ready to go...