It could be in the ignition switch. The bypass circuit is provided through the switch to the coil eliminating the ballast resistor for 12 volt starting (Ign 2). When you release the starter it switches over to the ign side of the switch (Ign 1). If there is a problem internally it may not get power to stay running.
Fred

1956 3100 Pickup/Red/350/3sp OD/PS/Disc Brakes
1957 Bel Air Sport Coupe/Red/355/TH350/PS/Disc Brakes
2017 Silverado LT Single Cab SB/Black/5.3/6 Speed Trans
1947 Willys CJ2A w/F-Head engine