Five days later, I caught up and returned to the Kingfisher where my Companions waited for me outside the World of Tree Spirit's Star System.
Since I managed to fool our pursuers about which general direction to take, I took my time hunting down the smaller Space Vessels they sent out. I made sure to be arbitrary on picking targets to throw them off which way they should really look for.
And given that the Serpent Knight died after he sent out his order to go down a layer, their main Space Vessel went there.