Now unfortunately, as per the rules on Removed from Game, when Hallowed Mirror leaves play, the 3 copies of Soothing Melody are rendered totally inaccessible, since a card removed from game "has no further interaction with the game in any manner for the duration of its removal." However, there's an easy workaround for this.
Just buy 5 extra copies of Before the Black Throne. Each mythos pack comes with 3 extra copies of Soothing Melody, which would give us a total of 15 backup bonded cards. Remember, we don't have to find the 3 specific copies that came with this one specific card, just 3 bonded cards total. This should be more than enough to fuel whatever recursion games we feel like playing with Hallowed Mirror for an entire scenario.
Of course, if any 3 copies of Soothing Melody will do, why not just proxy them? It's definitely cheaper than buying whole mythos packs, and works just as well for filling out our collection with all the copies of Soothing Melody that we could possibly need.
But then, if we're not bothering to get extra copies of Hallowed Mirror, why even bother proxying? We only have 1 copy of Hallowed Mirror, so only 3 copies of Soothing Melody will ever be in use at a time. There's nothing stopping us from just pretending that, once they leave play those 3 copies of Soothing Melody have been replaced by 3 new proxies that happen to have been made with the exact same cards.
Of course, skipping all this, we can always just realize that if a rules discussion can be rendered moot by just buying 5 extra copies of a mythos pack that you then don't even have to bother opening, that maybe "Hallowed Mirror leaving play means you can't get back the 3 copies of Soothing Melody if you play it again" was never how that was intended to work in the first place.