Green’s Dictionary of Slang

(US Und.) the bottom row of cells in a prison block.

1950		Goldin et al. DAUL 71/1: Flats. [...] 2. (P) The ground floor of any cell-block.
1953		F. Paley Rumble on the Docks (1955) 316: For God’s sake don’t [...] put him in the flats where he wouldn’t last an hour!
1980		(con. 1940s–60s) H. Huncke ‘Alvarez’ in Eve. Sun Turned Crimson (1980) in Huncke Reader (1998) 182: It is requited that the prisoners gather out on the flats and remain there until it is time for the midday meal.
1992		Bentley & Corbett Prison Sl. 7: Flag The bottom or ground floor row of cells in a cellhouse. [...] (Archaic: flats).