A spinor field in classical differential geometry is defined as a section of a spinor bundle, which is by definition the associated bundle constructed from the spinor representation and the orthonormal frame bundle.

When transforming from coordinates $(t,x,y,z)$ to $(t',x',y',z')$, related by a rotation of angle $\theta$ around the $x,y$ axes, I understand that the induced orthonormal frames undergo a rotation of $\theta$, so vector coordinates undergo a rotation of $-\theta$ while the spinor field coordinates changes by a rotation of $\theta/2$.

However, how would one express a spinor field in coordinates $(t',x',y',z')$ if the induced frame is not orthonormal?

For instance, consider changing from $(t,x,y,z)$ to $(t'=t,x'=x+y,y'=y,z'=z)$. Given a spinor $s=(1,0)$ located at spacetime point $P=(0,0,1,0)$ in the original coordinates, how should $s$ be expressed in the new coordinates?