An SA 3 speed hub is made so if you disconnect the cable it should be in 3rd. When you pull on the cable the tension will shift to 2nd and then 1rst. Try disconnecting the chain to see if it stays in 3rd, as it should. If it does you cable is probably too tight.
There are a few things to try.
1 Remove the cable and apply a little grease so it slides smoothly if it does not now.
2 You can inject a few ccs of oil into the hub through the drive side axle. This has made my 3 speeds shift and run quietly. When it becomes noisy again I put a little more in as it leaks out over time. I use synthetic 5/20 motor oil but there are differing opinions on this.
3. I would do this first. Disconnect the cable from the rod which comes out of the axle on the right. Screw the rod in all the way until tight and then back of a 1/2 turn or even a little more so the chain bends in the direction of the cable. Reattach the cable to the rod. The cable is taut enough when the smooth cylindrical part just below the chain attachment barely protrudes past the axle when you shift from 3rd to 2nd. This process is the standard SA adjustment.
Not being patient or mechanically gifted if these three do not work, I would take it to a shop that is familiar with SA hubs, although there are YouTubes available that discussing taking the hub apart, replacing broken or worn bits and reassembling.
