We study flatness of multi-input control-affine systems. We give a geometric characterization of systems that become static feedback linearizable after an invertible one-fold prolongation of a suitably chosen control. They form a particular class of flat systems. Namely, they are of differential weight n + m + 1, where n is the dimension of the state-space and m is the number of controls. We propose conditions (verifiable by differentiation and algebraic operations) describing that class and provide a system of PDEs giving all minimal flat outputs. We illustrate our results by several examples, in particular, an example of the quadrotor helicopter.