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The phrase "a second building" is correct and usable in written English.
You can use it when referring to an additional structure that is the second in a sequence or set.
Example: "The city plans to construct a second building next to the existing one to accommodate more offices."
Alternatives: "another building" or "a second structure".
Exact(60)
A second building was bought in 2005 using tax credits.
A second building with 70,000 square feet is planned.
That was a second building block of the governor's victory.
Half of a second building houses a cold storage morgue for body remains.
A second building has an oversize two-car garage, a workshop and an unfinished guest apartment.
A second building that stored food caught fire and was partly destroyed.
Behind a dingy courtyard was a second building with five more apartments for students.
A second building is being marketed, although construction has not started.
Next to NanoFab 300 South, a second building is already going up: NanoFab 300 North.
In June, the city Planning Board approved a second building, which will include stores, offices and three floors of condominiums.
A second building for secondary students is being refitted in what used to be Wembley town hall.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com