I zoomed in on the photo on a 55" HD monitor, it's hard to tell what's going on.
I'm guessing the red wire back to the motor is a safety feature.
But you really should track down the schematic and have a voltmeter on hand.

 

If both black and red go to the motor, the motor must be one that can be converted to 220v? The motor label should show you whether that is possible and may have a wiring diagram on it for the conversion.

I am not a electrician but this is what I would do: If the switch still works, with it unplugged, test the continuity of the black terminal coming into the switch to each of the red and black terminals going to the motor. If both show continuity with the switch on (and none with it off), then I think that you should be able to connect them both to the hot out on the switch you bought with a pigtail. The neutral (white) wires would be hooked to the neutral in and out on the switch as shown in the switch's instructions.

 

One more thought. In your picture, the wire nut on the ground wires is obscuring what looks like may be another terminal that the black wire is also hooked too? If that is correct, it is possible that the red to the motor will have continuity with one terminal and the black to the other. Assuming that the motor can be switched to 220v, I think this would allow you to change the cord and plug to 220v with 2 hot wires with one going to the terminal hidden by the wire nut and the other to the one we can see. In the current configuration, when the switch is on, the inbound hot wire makes both red and black to the motor hot with the same 120v.

 

There might be a wiring diagram on the motor showing wiring for 110 or 220. I would look at that and see which wire should be neutral (white) and which should be hot (Black). Other than that you are a test pilot.

 

Based upon the picture, it looks (to me) like the black and red are both hooked to hot on the switch and white currently goes directly to the motor unswitch. I would assume that the motor has both neutral and 2 hot wires but that is just a guess based upon what we can see.

 

Thanks for replying. Here's a diagram that shows how it's wired.

I think maybe it the copper coil is a start winding as shown in the diagram below.

(diagram from How to Connect a Capacitor to an AC Motor | eMotors Direct)

The red and black would attach to the start and run capacitors.

I think what I might do is attach the new switch to hot and neutral and then run the hot into the old box where the white arrow is in the picture.

Alternatively, maybe I just use a thin piece of wood for the back of the old box, mount it on the table saw and stop thinking about electricity.

 

Oh I see better now. The terminals with black in and out is actually the switch and the other must be a start relay. I agree that your proposed schematic makes sense.

 

First, the red wire does not mean it can be wired for 240. My drill press has bk, rd, gn cord. Something used overseas. If I understand you correctly, the switch has a single black in and a single black out. Just put these wires on the switch, and stop there. I can not tell what the black and red come out of due to the dirt. Clean everything before replacement. If you clean it up and move the wirenut holding the neutral out of the way, post a new picture, it might help.

 

Start and run capacitors, I assumed a blade-brake.
Well anyway you could jumper the live black to the lead to the motor for a quick test.
Use a fused powerstrip for a on-off switch.

Hopefully you don't need to find a OEM part.

 

I am pretty sure that the coil is a motor starting relay. I think that they are pretty common on compressor motors and I think that they are in lieu of a centrifugal switch. Pay attention to how it was originally oriented because I think that they may have to be oriented a certain way to work properly. I think they use gravity to turn on the start winding and as it draws the rated amps, the coil causes the relay to switch to run mode and turn off the start winding. Check out this video to see an explanation of how they work.

BTW, Even though it will work properly if you just run the hot through the switch, I would also run the (white) neutral through the new switch, as recommended in the new switch's wiring instructions. I seem to remember reading somewhere that this is a safer configuration because it totally isolates the saw if there is some sort of wiring fault in the house where the neutral is compromised. Something like that?

 

One rule for electrical installation is NEVER switch the neutral. There were switched neutrals many many years ago, and way back the neutral may have even had over current protection, like a fuse, but that is no longer allowed due to the dangers it creates. If a fault in your shop goes through the neutral to the saw, your shop is not wired correctly.

 

One rule for electrical installation is NEVER switch the neutral. There were switched neutrals many many years ago, and way back the neutral may have even had over current protection, like a fuse, but that is no longer allowed due to the dangers it creates. If a fault in your shop goes through the neutral to the saw, your shop is not wired correctly.

Is that just in the wiring installed in the building? The wiring diagram from the switch that he linked to shows the neutral hooked up to the switch.

 

A inductor in the starter switch, new one on me, however, I did checked out some videos and I work with inductors.
I'm thinking now that coil is balancing the motor's load, in any case you could simply move it to a new location.
I'll ask my nephew, he's a lead electrical engineer for a large firm.

EDIT - smoothing inductor
Good Guess Danger

 

RED IS LOAD BLACK IS LINE[ POWER SUPPLY RED IS TO LOAD, THE SAW

 

I do not think so in the case. The red and black are both hot to the motor. One is for the start winding the other the run winding. The coil (with black in and black and red out) is a relay that turns off the start winding once it is running.