This C.C. Pierce shot of Poundcake Hill seems to support the identification as being the jail.
Los Angeles County Court House and Jail on Poundcake Hill, ca.1895
Photograph of exterior Los Angeles County Court House and Jail on Poundcake Hill, ca.1895. Located on the south side of Temple Street between Broadway and Spring Street, on Poundcake Hill. Broad stairs approach the building from the corner. A stone retaining wall lines the sidewalk on the Court House side. About 10 horse-drawn carriages are parked on the street in front of the building. Utility poles and lines are visible. A flag flies from the uppermost clock tower of the building.
USC digital archive/Title Insurance and Trust, and C.C. Pierce Photography Collection, 1860-1960
Here you go, I think this shows us what you're talking about...I think we can see the high school has been moved and yet the courthouse hasn't yet been built. The small building we think may be the jail is up and can be seen just to the left of the distinctive roofline of St. Athanasius. At the base of this small building you can clearly see the dirt of Poundcake Hill and, in fact, I think the building may actually still be under construction.
Main, Spring, Temple Sts., ca.1887
Summary: The junction of Main, Spring and Temple Streets circa 1887, showing the Temple Block, Downey Block and County Jail.
LAPL00014004
And lastly, and perhaps least compelling because of the resolution, I think you can see the 'jail' building showing as a dark rectangle in this 1887 aerial on an otherwise vacant Poundcake Hill...
Aerial of Los Angeles, 1887
Aerial photo of Los Angeles on June 27, 1887 taken from a balloon. North is to the left. Note the farmland south of Second Street and east of Main Street to the Los Angeles River. The town's population was a little over 20,000 at this time. The circular form of the Plaza is visible to the center left. Of particular note, Poundcake Hill (look for the stub of Court Street pointing directly at it) is vacated for the coming construction of the new county courthouse which will be dedicated next year, in 1888, while the high school building has been moved and is visible over on Fort Moore Hill just above California Avenue.
waterandpower.org
annotated view...
Aerial of Los Angeles, 1887, annotated
I've always liked this shot looking south down Broadway in 1902. The camera is in or on the courthouse and I think we may be looking down on the roof of the county jail building right here in the left foreground.
Los Angeles, 1902
"Downtown Los Angeles of 51 years ago–an era when a four-story building ranked as a skyscraper–is illustrated in a photograph received yesterday by The Times.
Mrs. Glenn Randolph Schreiner of Indianapolis, sent it after reading a copy of the Times Midwinter Edition.
“Looking at the pictures of your beautiful modern buildings,” Mrs. Schreiner wrote,”I remembered a picture mailed to me years ago by an uncle who was working in your city at the time.” The uncle, Sloan Randolph, now lives in Vashon, Wash.
The picture, taken from the old Courthouse Hill, is a view looking southward down Broadway. Near its center is the tower of the old City Hall, completed in 1888. The 183-foot tower looked down on many large residences with lawns and trees on nearby Bunker Hill. The tower later was cracked by the 1918 quake which badly damaged San Jacinto and Hemet, and it was lowered by removal of a 76-foot section.
Long-time residents can spot the Nadeau Hotel, which stood on the site of the present Times Building; the Tally-Ho Stables, just north of 1st St. on the west side of Broadway, and the Central Jail, still in use on 1st St. between Broadway and Hill St.
The city has multiplied in population about 20 times since this picture was made in 1902. The census listed a figure of 102,479 in 1900.
The City Hall, the Nadeau and the Tally-Ho are gone today but some of the store and office buildings shown on Boardway and Spring St. are still in use.
The Times Midwinter Edition was published every New Year’s Day from 1885 to 1954. The edition, with many large photos, promoted Southern California."
Los Angeles Times archives/Los Angeles Times Midwinter edition 1953