Below is a scan from the 1929-54 Chevy parts book showing the solenoid starter. Maybe one of the experts on navigating these parts books can help with the question "Can a solenoid be added to a stomp starter?" If I were to guess the 4 mounting bolts for the solenoid would not play well in place of the 2 mounting bolts for the stomp switch. By the way, the solenoid is mounting backwards in this drawing.
'57 GMC 102, Original 347 V8, HydraMatic, 3.08 rear gear, added A/C, disk front brakes, HEI, AFB carb, '98 Honda Black Currant paint. T-boned and totaled 10/12 '52 GMC 152 Stake Bed, Original 228, SM420, added A/C, HEI, disk front brakes, '67 Chev 3.55 rear gear. Gets used as a real truck.