I use a Minify plugin to compress JS and CSS fi...

I recommend Better WP Minify, you just need enable Friendly Minify URL (from advanced tab) and that’s all!

You can also enable Rename Plugins from HMWP settings page or change cache directory by BWPM (create those directories manually and set permission to 777). For example change wp-content/cache/minify to /cache/m/.

For other plugins, it became even harder so we just suggest using one of the recommended plugins. Anyway, you should fix the problem by adding two (or more rules) to “Replace URLs” (one for each minified URL). You need to find old URLs so first DISABLE HMWP, make sure pretty URLs is disabled in minify plugin and then find minified JS and CSS URLs in the source code. Remove site address from those lines and use Replace URLs to replace them with new URLs (e.g. min_styles.css or min_scripts_js). Flush the cache (if you have) to see the result.

As an example for scripts, add this rule: (Note that first part always starts with wp-content not http://)

wp-content/plugins/minify/min/?f=wp-includes/js/jquery/jquery.js,wp-content/plugins/oqey-gallery/js/oqey-js-drag-iphone.js,wp-content/uploads/2012/04/orphus.js&m=1376074032 =>min_scripts_js

PS:
- Minification is a complex task. Sometimes even without HMWP it fails (because of other plugins or theme) so before making a support request make sure everything works well if you disable HMWP.
- This was tested with wp-minify and Better WP Minify (Read here, too)
- If you enabled new theme or plugin you need to check your source code again and change old URLs.
- As stated here, WP Rocket minification doesn’t work with Hide My WP.