Things may well have changed, but the old rule on adjusting the cable tension on 3-speeds was to place the bike in 2nd gear, then sight through the hole in the screw-on cylinder the cable runs into at the hub. When the cable is properly adjusted, the end of the rod that runs into the hub and shifts the gear should be flush with the end of the axle.
You Sturmey-Archer riders help me here. This is a 50 year old memory. Things may have changed. My memory may not be perfect. But it does sound like the OPs cable tension is not correct. (Probably the c able tension is a touch too loose and it is trying to slip into 2nd gear.)
Ben