Instant Queue Solves the Last Great iPod Dilemma
Hence: Instant Queue to the rescue! Instant Queue is a bare-bones solution to the "queueing problem." With Instant Queue, you can select one or more songs to be played next. And...well, that's it, really. Instant Queue will remain running in the background thanks to multitasking, and will automatically play the songs in your queue in order. Note that you can't import playlists or save queues.
I don't know if the "queueing problem" really warrants an entire app, but nevertheless—whenever I run into it, it's like an itch begging to be scratched. To continue the analogy, Instant Queue is a simple, efficient backscratcher. It doesn't try to replicate all of the Music app's features, though it mimics its appearance. Instead, it focuses on solving a single, irritating problem. And solve it it does.
If you, too, suffer from the "queueing problem," check out Instant Queue on the App Store.