When you need to add a teammate to multiple projects, doing them one at after the other can take up a lot of your time that you don't have.
You'll be happy to know, you can easily add them to multiple projects in one shot. All you need to do is be (or contact) the Company Administrator and have them add the user through the Company Settings page.
Once you click Add to Project, the list of projects available will load, you can use the checkboxes to add to Project or designate the user as Project Organizer.
You can only assign a user as Organizer if you are an Organizer for that same project, otherwise, you will only be able to add it as a team member. (see some checkboxes for the column Project Organizer are disabled)