You should be able to just customize the "upgrade.xsl" file within you extracted files. Then make sure that in the manifest.xsf you have your upgrade versions set to the minumum previous version and the new version of the form. Your customized upgrade.xsl should be applied when the old forms are opened in the new form template.

Since upgrade.xsl is just standard xsl, you can do what you want in there.

I was wondering if I could use a custom custom template version upgrade that
performs the built-in upgrade features, but does some work on the side, like
change some field values and/or call a webservice?
If so, what's the method I can call to perform the built-in upgrade function
(s)?

Thanks in advance.