Shear-driven flows over superhydrophobic surfaces formed of closely spaced circular bubbles are characterized by giant longitudinal slip lengths, viz., large compared with the periodicity (Schnitzer, Phys Rev Fluids 1(5):052101, 2016). This hints towards a strong superhydrophobic effect in the concomitant scenario of pressure-driven flow between two such surfaces, particularly for non-wide channels where bubble-to-bubble pitch and bubble radius are commensurate with channel width. We show here that such pressure-driven flows can be analyzed asymptotically and in closed form based on the smallness of the gaps separating the bubbles relative to the channel width (and bubble radius). We find that the flow adopts an unconventional plug profile away from the inter-bubble gaps, with the uniform velocity being asymptotically larger than the corresponding Poiseuille scale. For a given solid fraction and channel width, the net volumetric flux is maximized when the length of each semi-circular bubble-liquid interface is equal to the channel width. The plug flow identified herein cannot be obtained via a naive implementation of a Navier condition, which is indeed inapplicable for non-wide channels.