I don’t know.
I don’t think the brick building with the awnings would show up so “big” in the windows based on the distance it is likely from the West Wing judging from this aerial view:
I was thinking that the photographic equipment used to capture the image might have something to do with the depth issue (unique lenses) but- the trees seem to be an even bigger problem in supporting my guess, which isn’t really a guess... just taking the existing/surrounding structures into consideration. There is the green canopy however that lines up about right, granted the depth of field seems amiss.
Factoid slide- I believe that the press room was constructed where the WH swimming pool used to be located. FDR had the pool put in to help with his polio rehab exercises.
Interesting image/view. I see JFK initials in the www address. If it’s from the 60’s, some of those trees could have been removed considering it’s 60 years later.