He could have had workmen transported in covered trucks and not told who they were working for or where the digging was being done. In the old days money could buy silence a lot easier than now days.
There were a great many people out of work here during the depression era so finding a labor force wouldn't be hard.
Some old mansions up north have been found to have smugglers tunnels left over from colonial times. Some tunnels ran for many miles from the town down to caves on the shoreline.
Similar tunnels were constructed to hide out escaped slaves on the underground railroad.
There is a private subway line under New York City, built by the very wealthy to move about the city, or leave town for vacations, without being mobbed by news reporters, much like the Paparazzi of today.
A friend of the family owned a hillside which had an entrance to a huge tunnel complex built during the Cold War. I'd explored the original caves there before the complex was built.
The owner of the major portion of the caves sealed off the opening on my friends side of the property, leaving only a few yards of cave there. The rest of the cave system was turned into a huge fallout shelter with hospital dormitories and even class rooms. There were accomodations for several thousands of people there.
Last I heard the place had been condemned, but its still sealed and guarded. I suspect that since at most only a small fraction of the local population could take sheleter there that its still operational but only for a select few should it ever be needed.