SO to clarify, you can jump from the red fused wire to the wire that goes to the small terminal on the solenoid and it cranks, correct? It could be that the wrong switch was installed yes. One would need model numbers from below the seat and part number bought to clarify. You could also have an issue with a clutch or PTO safety switch as well.

I second what Tom said on the safety switches. Could be the seat switch or neutral safety switch too. Also check the connections from the bat + to the solenoid and + wire from the solenoid to the battery and make sure it's tight and clean, don't forget to check the ground(s) too.

Is it possible the engine is locked up for any of several possible causes and everything is working electrically but the engine can't turn? Just guessing because the description of what happened is so sketchy; like, why did you replace the ignition switch in the first place, and was the engine starting okay before that?