Tell TYPO3 extension to use different template using Typoscript

I'm trying to get an extension/plugin on a page to use a different template from the one its hardcoded with.

I made an html template stored in: typo3conf/ext/myextension/pi1/new_extension_template.html

I made a typoscript template object in the root of my site (so it would definitely get picked up). In it, I have defined the following simple Typoscript (names changed for demonstration)

plugin.tx_myextension_pi1 {
     templateFile = typo3conf/ext/myextension/pi1/new_extension_template.html
}

However when I load the page containing the plugin, I get a "no typoscript template found" error on that black TYPO3 error page.

I have tried a bunch of alternative ways in case the syntax was wrong but the above worked for me when working with the tt_news plugin.

E.g I have tried these without success too:

templateFile.file = typo3conf/ext/myextension/pi1/new_extension_template.html
...
tempfile.template_file = typo3conf/ext/myextension/pi1/new_extension_template.html

Any ideas what I'm doing wrong? Can html templates stored in typo3conf be called via typoscript stored outside the plugin?

I'm thinking perhaps html template files would have to be stored in fileadmin/plugin_templates/ for this to work.