Thermostat C-wire replacement using a wall socket

My second stab at answering this:

Since this is not a heat pump, I now see more clearly you have a photo that shows your blue wire connected to the coil of a contactor in the unit that is separate from the contactor controlled by the white wire but also with heavy supply wires coming into it. Judging from the large size of the wires fed into this contactor, I believe this was originally meant to be control for stage 2 electric heat. However, given the small gauge white wire that is connected to the load side now, I cannot say what has been connected in place of stage 2. I can say that unless that small white wire coming off that contactor is separately fused, then it is probably a fire hazard. I also see in the same photo that you have a lot of bare wire exposed on that heavy conductor.

I eventually got access to the other end and discovered that Daniel was indeed correct -- these were unwired.