[See also my update to this post.]
Yes, there once actually was a hospital atop Hospital Hill. The Leander M. Haskins Hospital opened for patients in 1906. It had 10 beds and, according to one report I found, treated 90 patients in a year. By comparison, the larger Addison Gilbert Hospital in neighboring Gloucester had 30 beds at the time. Those who live in Rockport today know that Hospital Hill is now bare, but I can find no reference to when the hospital stopped operating or what became of it. If anyone knows, please add a comment below.
As for Leander Haskins, the man for whom the hospital was named, he was, at least for a time, in the business of manufacturing isinglass, a substance made from the air bladders of fish and used in making glue, refining wines and liquors, and other purposes.
He was also someone who was active in town affairs. When the Sandy Bay Yacht Club was organized in 1885, he was its first commodore. When the town in 1903 needed a new library, he was the person who negotiated with Andrew Carnegie for the funding to build it.
A native of Rockport, Haskins graduated from Dartmouth College in 1862. A book about his class at Dartmouth, published in 1884, said this about him:
Leander Miller Haskins, son of Moses and Betsey D. Haskins, was born at Eockport, Mass., June 20, 1842. His father was a mariner. He fitted at Andover, Mass., entered the Scientific School in the spring term of 1860, and continued through the course.
After graduating he continued the study of engineering and surveying in Boston ; taught in the winter of 1862-3; went to New Orleans in May, 1863, and joined the 19th Army Corps, as Commissary Chief Clerk, stationed at Port Hudson and Carrol ton ; discharged by reason of sickness in September, 1863 ; appointed clerk in the Navy Department in December, 1863; resigned in April, 1866; continued the study of engineering in Boston until October, 1866, when he was reappointed a clerk in the Navy Department; resigned in October, 1868, and entered into a partnership with his brother Moses W. Haskins, in the wholesale fish and oil business ; in November, 1879 he engaged in the manufacture of isinglass, and so continues.
This postcard is postmarked 1907. No publisher is identified.