If a replacement solenoid hasn't fixed it then it's one of three things:
1. Faulty starter motor assembly
2. The red/light blue wire which connects to the starter relay is receiving 12 volts when it shouldn't be
3. The red wire from starter relay to starter motor assembly is receiving 12 volts when it shouldn't be.
You won't know which one of those it is unless you do some voltage checks using either a multimeter (DC voltage range) or a 12 volt test lamp. You won't want the starter motor to be cranking while you're checking voltages, so you'll need to disconnect the red wire which runs from the starter relay to the starter solenoid (the starter solenoid is physically attached to the starter motor body) - disconnect whichever end of the wire is easiest to access.
Put the gearshift in Neutral if it's a manual, or in Park or Neutral if it's an automatic, then switch the ignition to the Run position - that's the position where the gauges come to life and the dashboard warning lamps illuminate.
Then check the voltage at the red/light blue wire on the starter relay - it should be at 0 volts, so if you find 12 volts then that's the problem, and it will likely be a short-circuit somewhere along that wire.
If you have 0 volts on that wire then check the voltage at the terminal for the red wire on the starter relay - that's the red wire which runs from the relay to the starter solenoid. That should also be at 0 volts, and if you find 12 volts instead then that's a fault, and what happens next depends on which end of the wire you disconnected. If you disconnected it at the relay then that would indicate a faulty relay, but that's unlikely since it's a new one. If you disconnected it at the starter solenoid, then disconnect it at the relay end also and check the voltage again - if the relay terminal has now dropped from 12 volts to 0 volts but the disconnected wire still has 12 volts then there's a short-circuit somewhere along that length of wire.
Finally, if the relay checks out ok and you have 0 volts on the red/light blue wire and 0 volts on the red wire which runs from starter relay to starter solenoid, then that would be pointing towards a faulty starter motor assembly.
Let me know if you don't follow any of that.
By the way, I don't think it will be the ignition switch because even if the switch's Start contacts were faulty and sending a constant 12 volts to the starter relay coil, the Park/Neutral Position Switch would interrupt the circuit on an automatic and prevent it from cranking, or the Clutch Pedal Position switch would do the same on a manual. Therefore I reckon it's not the switch.