I have to override Widget Search template to display some additional informations.
The name of the template is asset--widget-search.tpl.php
I duplicate it in my theme folder and customize it.

On initial load, all is fine, my customization are visible in the library.

Then I use a form to add an image for example, and Save the Asset, the library reload.
--> On reload, my customized template is gone !

I made a little test and write some letters in the original template, diretly in the asset/modules/asset_widget/theme/ folder, and add a new Asset again.
--> I can see that this template is used just after a new asset is saved.

So it' seems that the overridden template is used on initial load, then after that it's the original one that is used after adding a new Asset...