Forget about messing around with apps. There’s a powerful voicemail handling system built into nearly every big carrier today — and that system almost always includes a system for recording a voice message and sending it directly without ringing the target phone. I will walk you through the process of doing this on Straight Talk (since that happens to be my carrier) but most carriers will have similar systems.
Later, you can access the recording like you would any other voice mail message. If you desire, export voice mail messages as audio files.

If you never recorded one and want to record or change a voicemail greeting, you've come to the right place. The process is very simple. You'll just have to click the green phone icon, the same place you make calls, and navigate to the voicemail screen. On the voicemail screen, you'll see a greetings option on the top left.
If you’ve already created a custom voicemail on your iPhone, you can tap Play to see how your voicemail sounds. If you’re interested in creating a new message, tap the Custom button. From here, you can start recording your message by tapping Record and tapping Stop whenever you’re finished. Once you’re finished recording, tap on Play to make sure the message comes across as you’d like and hit Save (in the upper right) to save it as your voicemail.