How to add script right after ?

Fairly certain this is not possible without editing the actual theme header.php file. The function wp_head() does in-fact include other scripts, not sure you get a choice to add a script right above everything else and to be honest I don’t see why you would need too.

Your constraints seem rather strict, I’m curious why you can’t edit the header.php file. It’s not like updating WordPress will overwrite your changes.

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