I'm confused. You can start it anytime by jumping the solenoid. If that is the case then it could be the ignition switch. It could be one of the safety switches like if you don't stay seated or the other ones, I don't remember them because I bypassed them. I would start at the ignition switch and see if you have power coming out in start position and if you do go down the line and check everything else in between. You should be able to get a electrical schematic for the mower online.
It sounds like a bad starter and solenoid hearing the rest of the explanation. I would change them and clean all of the connections.
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