I also immediately thought that this machine would be a perfect candidate for a turboshaft engine. A little googling showed I was only about 60 years late with that idea. Westland helicopters built a turbine variant in 1958 as the Wessex.
Production of the H-34 type ended in 1970 but the next year Sikorsky started doing conversions to a Pratt & Whitney Canada PT-6-3 Twin-Pac® Turbine engine. This was a big step up in reliability and safety, though in my opinion, the longer nose didn't add to its looks.
Twin-Pak Turbine powered H58-T