Yes! You can sign into multiple teams/organizations simultaneously and switch back and forth between them as necessary.
When you are signed into multiple teams they are visible to you in a sidebar so that you can see if you have pending/missed notifications in any of your teams.
In order to do this, navigate to the top left where you see your current organizations photo and click the photo.
This will bring you to a screen where you will see blue text that says "+ Add Team", simply click that text and HighSide will look as if you are loading it up for the very first time!
From here you can use your secret key from a team you are already a part of or you can accept an invite to a new team with a brand new signup key.