Will the starter work if you jump from the Battery cable to the "S" terminal on the solenoid?
If it does, you probably need what is termed a "Hot Start Kit" which, basically, is a 12V-40 A relay, primary would be the "S" wire, and the secondary, a 14 Ga wire switched through the Relay to the "S" terminal on the solenoid.
GM's "better idea" (Way back when) was an 18 ga "S" wire and over the years, some of the strands in that wire apparently get frayed (especially in cars w/ tilt steering wheels) and with the new quarter wound solenoids, you don't provide enough amperage to make the electromagnetic switch in the solenoid work . . .
Went through this with my street rod a few years back . . Haven't had a starting problem since I applied this little device . .