Welcome to 2CarPros.
Yes, it sounds like the solenoid was wired wrong or it is bad even though it is new. I need to confirm exactly what was replaced. Was it the starter relay which is located on the right front inner fender or was it an actual solenoid on the starter itself? There are two different systems used on this vehicle, a Delco Remy and a Mitsubishi system. They are different in nature, so I have to be sure of what I am dealing with prior to making recommendations.
I have attached a picture of the relay on the fender. See picture 1. If the red wire and the red/light blue are switched, you are going to have constant power to the starter.
The second picture is an overall wiring schematic of the starting system. Since this started right after the repair was done, something is either put together wrong, or there is a short to power which is bypassing the ignition switch itself. Because this type of issue can be somewhat on the dangerous side (dead short can cause a fire) you may want to return the vehicle to the repair shop for them to recheck their work. Also, until it is fixed, even if connecting the battery doesn't start the vehicle, leave it disconnected until checked and don't park it in a garage connected.
Please let me know if I can help you further.
Take care,
Joe
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Thursday, May 23rd, 2019 AT 7:36 PM