I’ve never used a non-contact tester, so I can’t really say. You really should get a digital DMM. You can get one for less than $20 on Amazon.
I am assuming one side of the switch goes upstream to the circuit breaker box, and connects to a circuit breaker. The other wire (short red wire) goes to the hot side of the outlet.
What is the rating of the circuit breaker? 15 A or 20 A?
Is there a ground wire anywhere in the box?
So thinking about this a bit more, here’s what I would do:
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Cut power.
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Unplug anything from downstream outlets.
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Remove switch, outlet, and short red wire. Discard all of these.
So now there should be four wires in the box. Let’s call them A, B, C, and D:
A: This is the wire going to the switch. I am not sure what color it is.
B: Black wire currently going to the left side of the outlet.
C: White wire going to the right side of the outlet.
D: Black wire going to the right side of the outlet.
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Make sure no wires are touching other.
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Turn power back on.
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Turn knob on DMM to measure AC voltage (200 V scale or higher), and carefully make these voltage measurements:
A vs. B
A vs. C
A vs. D
B vs. C
B vs. D
C vs. D
Now, for the above measurements, one of them should be 120 VAC. These two wires are going “upstream” toward the circuit breaker box: one goes to a circuit breaker in the breaker box, while the other goes to the neutral/ground bus bar in the breaker box. I would assume one of the wires is A, and I would further assume A goes to a circuit breaker in the breaker box. But I would want to confirm this. To confirm this, measure the voltage between each of these two wires and a known ground: one of these wires will be 120 VAC vs. ground; this is the “hot” wire that goes back to a circuit breaker in the breaker box. It will connect to the switch, and the other side of the switch will connect to the hot/line side of the GFCI receptacle. The other wire will connect to the neutral/line side of the GFCI receptacle.
So now you have the other two wires. These go downstream, to other outlets or whatever. You need to determine which one is hot and which one is neutral.
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Cut power.
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Plug a long extension cord in to one of the downstream outlets.
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Rotate switch on DMM to resistance mode. Select the lowest range.
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Verify which wire is hot (short/narrow prong) and which is neutral (tall/wide prong).
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Put everything back together. For the short wire between the switch and outlet, just cut it from a roll of NM cable (“Romex”). I would use 12 AWG wire. But I suppose you can use 14 AWG if the circuit breaker is rated for 15 A.
Oh, and when you buy a new switch, get one rated for 20 A, not a 15 A, regardless of the circuit breaker rating.