I think this one actually now qualifies as a combo since any reshuffle triggers it, so I pulled it from the other thread.
This one was really kind of neat, although an extreme edge case. If you gain a Siren with a Sheepdog in hand and you are at the end of your draw pile you can gain it without trashing if the Siren ends up triggering a reshuffle.
I was wondering if it mattered whether you actually draw the Siren there, and it's easier than that. I just tested it online, and this works whenever the Sheepdog makes you shuffle the gained Siren into your deck--that's where the Siren loses track of itself. You don't have to draw the Siren from the Sheepdog effect, just shuffle the Siren into your deck with the Sheepdog effect.
From my rules document, there's these, that can move the Siren directly:
Bauble, Cargo Ship, City-state, Deliver, Gatekeeper, Hasty, Innovation, Insignia, Royal Seal, Rush, Sailor, Sleigh, Tiara, Tracker, Trappers’ Lodge, Travelling Fair, Watchtower, Way of the Seal
And these, that can cause the Siren to be shuffled in:
Band of Nomads, Sheepdog, Way of the Horse/Mole/Mouse/Otter/Owl/Pig (via Falconer, Sheepdog or Stowaway)