The original plan for Montauk was envisioned by Carl Fisher. He planned an 18-hole golf course, (Montauk Downs) and to make Montauk the "Miami Beach of the North", but he went belly-up and the Montauk Beach Development Corporation were the next owners of the property in 1926?. They, too, lost it all in the coming depression, but there were plans for a second hotel, golf course, etc.
This is the brochure for the second hotel:
Cover:
Inside:
And the golf course they planned was to be known as Hither Hills GC.
An article about the course:
This photo is from 1938, a mere 10 years after it was built. Looks like it too was a casualty of the depression. The hole corridors are still visible in a historicaerials.com photo from 1954 , but a smattering of houses had been built on them by 1960. Nothing resembling a golf course remains past 1969.
I had no luck in divining the architect of Hither Hills, but perhaps they used Captain H.C. Tippett, the man who built the Montauk Downs GC in 1927.
Here's a 1938 aerial of Montauk Downs (built in 1927), and it seems to be in running condition, unlike HH.