We have or going to have numerous users who will be able to access the site at different levels of the content tree, but the current link to the experience editor is just https://[domain]/?sc_mode=edit. I see where this is set in the core database, but is there any way to change it per user, or normally would a user simply be directed to the "highest" folder of the site they have permissions for? (We are having another issue with the experience editor showing an error when we access it for a specific page for a non-admin, but I think that may have to do with a coding issue on a layout).
Easiest way I can think of would be to create multiple copies of the button, change the target url in each link to your desired target. Then set the security on those buttons for the roles that should get that "version" of the button. So deny view on the roles that shouldn't see it, and allow the view for the role that should.
Once you add the users in the correct roles, they would get a button that is customized to their usage.