One of the most quoted Bacon’s caning cases, that of the expertly publicised Sue Olds which you instance, refers to caning on the hand.
The principal case I recall which specifically alleged caning of a girl on the bottom by the Headmaster concerned involved an initial refusal by the pupil to accept the cane. The practice of substituting a punishment perceived as more serious, in that case bottom instead of hand as the target of the cane, was by no means unusual under such circumstances in that era.