Using Recessed Lighting with a 6 pin connector instead of a 4 pin?
Just pigtail off to another 773-104

I would simply get another Wago 773-104 or equivalent connector and use a short length of 12AWG wire to pigtail the connector on your recessed lights to the new connector.  Repeat this for the neutral and ground, and you should be good to go!

As an aside -- one could pull the wire out of the connector, but it tends to nick the wire up, in addition to rendering the connector useless.