Possibly the Willard Branch and the Willard Hospital spur did connect. Archer, in A HISTORY OF THE LEHIGH VALLEY RAILROAD, on page 135 gives the following information:
The Willard Branch was origionally the Hayt's Corners, Ovid & Willard Railroad, built as NG and opened in 1883. The LV aquired the line when it took over the GI&S as the GI&S had leased the HCO&W several years prior. For a time the line was operated as a dual-gauge road. Now, Archer relates that the Willard Branch formed a bridge line between the old and new Ithaca-Geneva mainlines when the Seneca Lake bypass was routed through its Willard terminus. I find this statement to be a bit curious, but understand it as meaning that there was a possible connection between the Willard Branch and the Willard spur. Beats me. One can draw their own conclusion.
Had a post card of the Willard Hospital steam locomotive. It was either a 0-6-0 as BR&P mentions above or a 0-4-0, I honestly cannot recall as I sold the card several years ago. Do remember that the machine was lettered for the Willard Hospital.
Was going to walk the Willard Hospital grounds in order to follow the remains of the branch and spur. Never got to it, alas. When I finally determined to do the walk it was too late as I discovered that some of the grounds had been converted to what appears to be a high security prison.
You can't go home again- Thomas Wolfe