Real Thumbnail Generator allows you efficiently to regenerate your (existing) thumbnails in your media library. You can regenerate a single image or images in bulk. The plugin allows you to create a custom upload structure for thumbnails.
The regeneration of thumbnails can be up to 33% faster than other plugins due to chunked regeneration. Just install this plugin and it works with all your image and media files without any problems. Another helpful feature is, that unused image sizes can be detected and deleted, for example if you have switched the WordPress theme. It also supports WordPress Multisites.
In addition, you can set a custom upload structure the media settings. Here is a little example how the WordPress standard upload looks like:
wp-content/uploads/2016/11/myimage-150×150.jpg
… and after you set a custom upload structure it can look like:
wp-content/uploads/2016/11/thumbnails/medium/myimage.jpg