When I try below:

backup database not_existing_db to disk = 'E:\Backup\not_existing_db.bak'

SSMS shows two error messages:

Msg 911, Level 16, State 11, Line 22
Database 'not_existing_db' does not exist. Make sure that the name is entered correctly.

Msg 3013, Level 16, State 1, Line 22
BACKUP DATABASE is terminating abnormally.

When I try to encapsulate it into TRY...CATCH block like below:

begin try
    backup database not_existing_db to disk = 'E:\Backup\not_existing_db.bak'
end try
begin catch
    select ERROR_MESSAGE()
end catch

error_message() only shows second error which is not informative at all

I want to find a way to log both messages, so most informative error message (first) could be send by email to DBA

Please help to find out how to do that