Let $X=(X_t)_{t\ge 0}$ be a self-similar Markov process with values in the non-negative half line, such that the state $0$ is a trap. We present a necessary and sufficient condition for the existence of a self-similar recurrent extension of $X$ that leaves $0$ continuously. This condition is expressed in terms of the L\'evy process associated with $X$ by the Lamperti transformation.