- suggestions to remove, clean and tighten all cables are always good especially in these high current connections.
- sometimes a trick to check if it is the solenoid is to have one person be in car and turn the key and then hit the solenoid with a heavy tool like a hammer- not too hard. This can loosen it. Sometimes it is not moving its core due to rust and dirt or its actually too weak because of too much voltage drop (see 1*)
or another trick to see if it is the solenoid/cable is to bypass the ignition:
shorten the connection from the positive cable of the solenoid to the starter. This can be dangerous because there is no fuse on the starter wire and a short to ground (any metal part in the engine compartment ) can cause high current to melt your tool. Also this bypasses the safety in newer cars that when not in neutral or parking the starter will move the car.
Assuming the starter has a good connection to the battery you will bypass any relais, loose wires, ignition switches (My dad did this with a large screwdriver or other metal tool.
In most cases if you found out that it is the solenoid try to replace only that one if possible.