Two High Priests remained alive from the time of the First Temple, and they argued about this walking. (This sounds too improbable, and some versions of the text simply read "from the time of the Temple," omitting the word "first.") One said that he walked around the Altar "with his hand," and another that he walked "with his foot." Each gave a reason for his words.
In any case, after the service was done, the remaining blood was poured onto the western corner of the Outside Altar. That is just because when he exits the Holy, he meets this corner first. But for all other sacrifices, they poured the blood at the southern corner. All that blood flowed into the canal and further into the Kidron Valley, where it was sold to gardeners for fertilizer.
Art: The Lincolnshire Ox by George Stubbs