Samsung's 'S Voice' assistant app has improved considerably on the Galaxy S4. In fact, it's worth revisiting if you dismissed it on earlier devices due to slow performance or hit-and-miss speech recognition. On the other hand, if you're not using S Voice regularly, there's a quick, easy setting change you can perform to make the home key just a little more responsive.
By default, when you double-tap the home key, you'll launch straight into S Voice. That means when you press home, there's a very short pause while the phone waits for a second button press. To avoid this and have the home key immediately send you back to the home screen, you can disable the S Voice shortcut combo. To do this, load S Voice, then press the menu key and hit "Settings." Next, uncheck "Open via the home key." That's it -- with the shortcut combo disabled, the home button should respond immediately as opposed to with a very slight delay.