If you are concerned with multiple users being on the same machine and not signing out between sessions, you can force a sign-out before starting the OAuth process. To force a sign-out, send the user to the page https://orcid.org/userStatus.json?logUserOut=true.
We suggest hitting that page as part of a script on the page where user starts the OAuth process. A working example of this is at https://orcid.github.io/test/log-user-out-jsonp.html. View the page source or the project in Github to see how the sign-out is called as a script on that page.
The user is also signed out of their ORCID account if they visit the sign-out page at https://orcid.org/signout.