One idea is that they can make some of their work elements BE instruments. We know how to make, say, a music-box, or a "piano-roll" type of punched spaces (which can allow varying air-flow like wind instruments) -- so if they need to MILL their grains, they can design the mill to have a musical element - a pleasing harmonic background, "wasting" some of the energy generated, potentially, but in the service of improved morale.
With this baseline, others may improvise or share songs as they work.
You may have a few people whose main job is music/morale -- it's worth giving up one set of hands (to play the instrument) if it keeps 10 people working longer, or more in-synch, or more productively. More likely, I see it being a turn-based thing, so at the start, no one gets to be ONLY a bard (unless you want that for dramatic tension purposes).
Since I re-read the question and saw it's post-apocalyptic, how long do batteries last? Can they be hooked up to run existing players (whether vinyl, cassette, CD?)
It may be worth researching the Amish and other groups that are deliberately limiting their use of The Grid -- I believe I've read they have cell phones, kept in a central place for use in emergencies, for example.
Perhaps your society can have a resource library where one person per work-area checks out a media-player, media, and power (to avoid redundancies, and the isolation of headphones -- brings music back to being communal).