• Log out of the account you're logged into the Steam app with, do this from Settings > Sign Out
• Log into the Steam app with the next account.
• Follow the steps here for adding the mobile authenticator to this account.
• Repeat for each account.
The key thing is to add the phone number to your account from the app. If you try it from the browser or client, it will tell you it's going to remove the phone number from the other account. Via the app it doesn't do that.
After you're done, you can swipe left/right between the different accounts to get the codes for each account.

***
Don't try to add the phone number to your Account in steam.
DO Login with your Account on the Steam mobile App.
Go to Steam Guard.
Then Steam Guard settings.
Change From E-Mail Codes to Steam-Guard.
NOW the App will ask you to add a Phone number for the Verification SMS.
And Done.