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Booth used two stables on the day/night of the assassination. The first was the livery stable owned by James W. Pumphrey, from which he rented the bay mare upon which he later fled from Ford's Theatre after shooting Lincoln just after 10 pm on April 14th. Most accounts (Manhunt by James Swanson and American Brutus by Michael Kauffman stand out as the best) have Booth renting the horse around 12:30 pm and picking it up about three hours later. Alexander Gardner did photograph a livery stable as a site associated with the assassination after the fact, but it was the Howard Livery Stable between 6th and 7th Streets, likely not the same one from which the mare was rented. The photograph and relevant research can be located at https://civilwartalk.com/threads/livery-stables-vanished-america.123658/.
However, the stable you seem to reference in your message is the smaller brick building which was located about 100 feet behind Ford's, which was where Booth rented the space to keep his horse during the hours preceding the assassination when he went drinking at the Star Saloon to build up his courage for the act. At the exact time of the assassination, the horse was being held by a young stage hand (John "Peanuts" Burroughs) who then had the horse wrestled away from him by Booth as he fled through what remains known today as "Baptist Alley" (it was the site of an old baptist church before Ford's was built on the site). No photographs were taken of the stable, but the Washington Post did chase down the relevant information about the building in an article about 15 years back regarding the building falling into disrepair. See Stable Used by Booth Needs He ... - The Washington Post.
Closest contemporary photos of the spaces around the stable would be these: From behind Ford's less than 100 feet in front of the stable https://s-media-cache-ak0.pinimg.com/736x/ef/73/e5/ef73e557858cd7ecab17f9edec35ed27.jpg; and from between 9th and Tenth and F Street looking into the alley from the South https://s3.amazonaws.com/uploads.knightlab.com/storymapjs/9f7876575cd67e9aa4e72b8fb0bc7d7d/the-escape-route-of-john-wilk… Both were taken years if not decades following the assassination.
Your best bet to chase down specific information and narrow down your search assuming such a photo of the stable does exist would be to contact the National Park Service and the Ford's Theatre Society, who operate the reconstructed building as both an historic site and a modern theater. They would have a better sense of the photographs taken at the time and after around the theater (the Mathew Brady images come to mind, but all of those were taken in front of and inside the structure) and control the relevant images in their collections. The Theatre Society can be contacted at (202) 347-4833 and the Park Service unit in charge of the site can be reached at (202) 426-6924.
Hope this helps. Thanks for your inquiry to History Hub!
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