We give a new proof of former results by G. Zampieri and the author on extension of holomorphic functions from one side $\Omega$ of a real hypersurface $M$ of $\mathbb{C}^n$ in the presence of an analytic disc tangent to $M$, attached to $\bar\Omega$ but not to $M$. Our method enables us to weaken the regularity assumptions both for the hypersurface and the disc.