How can I add a switched circuit to an existing three-way circuit?
You mentioned the travelers were in a /2 cable.  I hope that's a typo; 3-ways require at least /3 cable, and NEC 300.3(B) requires all related wires to be in the same cable or conduit, to assure currents are equal and the rather intense dynamic EMFs of AC power cancel each other out.
First, on the 3-way circuit, don't mess with the travelers.  They are not useful unless you want byzantine and annoying light control.  I even recommend marking the two travelers with yellow tape on both ends so you know what you're dealing with.

The switched-hot wire, however, can be used as a source for a subordinate light circuit.
You wire it like a normal light circuit, except instead of &quot;supply hot&quot; you take &quot;tier 1 switched-hot&quot;.