  We show that if $E/\mathbb{Q}$ is an elliptic curve without complex
multiplication and for which there is a prime $q$ such that the image of
$\bar{\rho}_{E,q}$ is contained in the normaliser of a split Cartan subgroup of
$\rm{GL}_2(\mathbb{F}_q)$, then $\bar{\rho}_{E,p}$ surjects onto
$\rm{GL}_2(\mathbb{F}_p)$ for every prime $p>37$. This result complements a
previous result by the author. We also prove analogue results for certain
families of $\mathbb{Q}$-curves, building on results of Ellenberg (2004) and Le
Fourn (2016).
