We show that the $F$-signature of an $F$-finite local ring $R$ of characteristic $p\>0$ exists when $R$ is either the localization of an $\mathbf{N}$-graded ring at its irrelevant ideal or $\mathbf{Q}$-Gorenstein on its punctured spectrum. This extends results by Huneke, Leuschke, Yao and Singh and proves the existence of the $F$-signature in the cases where weak $F$-regularity is known to be equivalent to strong $F$-regularity.