As long as you are accessing the WordPress dashboard via HTTPS, new media items should be inserted into content using HTTPS in the associated media item URL. There is a bit more detail around that feature in this WordPress ticket: wp_get_attachment_url returns https when it should not.
That ticket describes how wp_get_attachment_url returns URL's that are protocol relative following an update in version 4.2. Following the related changes made editing any post and using wp_get_attachment_url to pull in URL's will use HTTPS.
Using relative paths for resources is another viable option, but since WordPress has various pieces of functionality that specify a full path, it seems easier to skip that option.
How to Correct
To correct any old references to HTTP URL's, I would recommend using the various tools available to execute a "search and replace" across all posts/content in your site's database. You would essentially be searching for
http://domain.com/.... and replacing it with
https://domain.com/.... My personal preference is interconnectit's serach and replace tool: https://interconnectit.com/products/search-and-replace-for-wordpress-databases/
However wp-cli is another solid option. Also you can go the plugin route with something like Velvet Blues.