As far as i know WordPress will not automatically recognize any of your templates if you put them into a subfolder. I guess furthermore that there is not hook to fix this. However you can search the core and try to modify this.
What I recommend is the following: Do not use the WordPress auto detection of template files it is quite clumsy anyway. In your index.php file use conditional tags in order to route WordPress correctly.
https://codex.wordpress.org/Conditional_Tags
This way your are 10 times more flexible regarding subfolders as well as code: You do not need to copy/paste code within the template files.
Use
get_template_part(..)
within your routing functions.
I have built hundreds of WordPress sites and I always do it this way. It is also recommended in some WordPress books.
Note: The loading mechanism of WP is shown within the following image: https://developer.wordpress.org/files/2014/10/wp-hierarchy.png
Update 2018: It is possible to put page templates which are selectable in the post edit screen into a /page-templates/ subfolder and they are detected automatically.