I do not know if this is the hospital in question, but since the report was written 31 Dec 1863 (which is right after the skirmish at or near Blain's Cross-Roads that occurred 16-19 Dec 1863) it is possibly the one.

OR, Ser I, Vol 31, pt. 1, pp. 664-665
...4 1/2 miles beyond Bean's Station. Here they found a rebel hospital with a number of wounded, too badly to be moved. They took 4 prisoners- part of the hospital attendants.
Report of Maj. Gen. John G. Parke, U S Army
Strawberry Plains, TN 31 Dec 1863

I ran across this report by thinking the hospital might have been in Rutledge.