That would surprise me. .308 would be my expectation, maybe .30-06.
Apparently the Utah Department of Corrections has been using Winchester 94s (as of Gilmore's execution in the 1970s) since the first decade of the 20th century.
Not surprising, really. 94s were a known entity, they were reliable, and it was likely that every guard who was handed one was familiar with the use and operation of a Winchester lever action.
It wasn't until the early 1970s that the Los Angeles Police Department ditched the .30-30 Winchester, one of which was in the trunk of each patrol sergeant's car.