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New solenoid for what?
Ok, great. Now, is this the starter solenoid on the starter itself or a solenoid/relay not on the starter itself?
Also, what engine came in this and does this have an automatic or manual transmission?
Did you double-check that the solenoid wiring was installed back on the correct terminals?
Ok, was this the reason that you replaced the relay/solenoid in the first place, because the starter wouldn't work?
If it was, does that relay click when you turn the key to start? If not, you might have a bad neutral safety switch on the transmission. Remove that relay and test the prongs in the socket for voltage when the key is turned from off to start. There should be one prong hot all the time and an additional hot when the key is turned to start.
Is this "faint single dull click" coming from that relay you just replaced? can you reach it or have someone else turn the key to start while you touch the relay and feel if it is what is clicking.
Ok, then voltage is reaching it. Is the battery fully charged up?
wait, you tried a jump start and it still won't crank over? I believe you have a bad starter! Unless the cable from the relay to the starter is loose, shorted.
Do you have a voltmeter or 12 volt test light in order to test for voltage at the starter?
If so or if not, you can usually purchase either for $5 or so almost anywhere. Then test for battery voltage (12 volts) at the terminal on the starter when the key is turned to start and be sure the vehicle is in park, with parking brake applied firmly.
If not repair or replace that wire from the relay to the starter. If it's just loose, tighten it up. If there is 12 volt there, remove the starter and replace it but first have it bench testd at the auto-parts store.
If the starter checks out good, then answer me this. Did the engine die when running? If so, check the motor oil level and see if the engine turns over by hand. Maybe the engine seized up but I doubt it or you would have probably said the engine made noise, smoked or died.
No, not ohmmeter, voltmeter, although most are multi-meters.