Are there any other iOS apps other than Mail which can send email?
The iOS version of Sparrow has just been released, and it contains this functionality. In the settings for email accounts you can set up aliases. You can also set up a custom signature and SMTP server for each alias:




When composing a message, by default it will be sent from your default alias. If you want to change this, tap the "From" in the top center and you will be able to select which alias you want this message to come from:


 

Sparrow is available in the App Store.