Recently, many existence results for the stochastic thin-film equation were established in the case of a quadratic mobility exponent $n=2$, in which the noise term $\partial_x(u^\frac{n}{2}\mathcal{W})$ becomes linear. In the case of a non-quadratic mobility exponent, results are only available in the situation that $n\ge \frac{8}{3}$ leaving the interval of mobility exponents $n\in (2,\frac{8}{3})$ untreated. In this article we resolve the current gap in the literature by presenting a proof, which works under the assumption $n\in (2,3)$, i.e., the regime of weak slippage. The key idea is to use that the $\log$-entropy dissipation coincides with the energy production due to the noise. To realize this idea, we approximate the stochastic thin-film equation by stochastic thin-film equations with inhomogeneous mobility functions, which behave like a higher power near $0$. As a consequence the approximate solutions are non-negative, which is vital to use the $\log$-entropy estimate.