You can add another account by setting up SMTP Accounts
. When you have an SMTP Account set up, it will show in this dropdown so you can choose which account to send messages from.
Also, as kevind stated (thanks for your response, Kevin!), Domain Aliases will show up here as well, but user aliases won't. The reason you can send from the domain alias is because it uses the users' mailbox to send from, whereas user aliases don't have an associated mailbox.