Let $C$ be a smooth projective algebraic curve over the complex numbers and let $V$ be a stable rank 2 vector bundle on $C$. Then consider the vector bundle $\mathcal{E}nd_0(V) $ of tracefree endomorphisms of $V$ and the map $h_V: H^0(C,\mathcal{E}nd_0(V)\otimes K_C) \to H^0(K_C^2)$ sending a twisted endomorphism $\phi$ to its determinant. Is this "restricted Hitchin map" dominant (or even surjective) or are there stable vector bundles for which this fails?
A sufficient condition for this is that $V$ is very stable. Then $h_V^{-1}(0)$ is a finite scheme and we can conclude that actually all fibers of $h_V$ are finite. By dimension counting (the source and the target space both have dimension $3g-3$.) $h_V$ is dominant. A necessary condition for the dominance of $h_V$ is the existence of a twisted endomorphism which is everywhere nonvanishing. This holds because we find $\phi \in H^0(\mathcal{E}nd_0(V)\otimes K_C)$ for which the differential of $h_V$ is surjective. In particular $(V,\phi)$ is a regular point for the classical Hitchin map. Therefore $\phi$ is everywhere non-vanishing (see Hitchin - "critical loci for Higgs bundles").
This question can be asked in much more generality for $SL_n$-Higgs bundles or $GL_n$-Higgs bundles i believe.
Edit: After Will Sawin's comment i realized that the question is not even trivial for $V = \mathcal{O}^2$! (which is not stable but let's not worry about this) By considering the derivative of the determinant a reformulation of my question in this case is: Are there $\phi_1,\phi_2,\phi_3 \in H^0(K_C)$ such that the map $H^0(K_C)\phi_1\oplus H^0(K_C)\phi_2 \oplus H^0(K_C)\phi_2 \to H^0(K_C^2)$ is surjective. This would be a substantial strengthening of Max Noether's theorem. For smooth quartic curves this is true because $H^0(K_C)$ is then 3-dimensional. On the other hand it fails for hyperelliptic curves as pointed out below.
Another interesting toymodels are semistable (or even unstable) vector bundles which are a direct sum of line bundles.
