The pigtail is the correct one, but do not assume that the switch is bad..the most likely problem is your a/c system is low on refrigerant and not allowing the high side pressure to get to the cut-in spec pressure of the low fan switch.
Look in the eye-sight and see if you have lots of bubbles..or better yet, put some gauges on the system and see what the high side pressures are.
All as you have done with the jumper is verified the elctrical system for low fan, minus the switch..you have not yet diagnosed the No Low Fan problem.

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A Dalton