Yes
The red wire that came with the kit (the one that's too short) is known as "single core shielded wire" (or cable).
It has the inner insulated wire (the "single core") next there is the braided wire (the "shield") and last is the outer insulation (the red pvc).
The single core is your positive (or "hot") lead that goes from the middle terminal of the 3-way switch to the "tip" terminal of the output jack.
The braided shield is also your negative (or "ground") which goes from the big lug (opposite the 3 positive terminals) of the 3-way switch to the "sleeve" terminal of the output jack.
NOTE: The photo shows a 3 terminal jack. Just ignore the middle one and look at the wire.
