Weinstein, in the paper where he defined the concept, gives a criterion, if $H^1(\Sigma)=0$, the characteristic line bundle comes with a distinguished orientation. In the same paper he also constructs an hypersurface in $\mathbb{R}^4$ which is not of contact type, by showing that the criterion is violated.

Many hypersurfaces are of contact type, as you remarked. Another nice example are *mechanical hypersurfaces*. These are hypersurfaces arising from hamiltonians $H=T+V$, where $T$ is the kinetic energy term $T=1/2|p|^2$, and $V$ is a potential depending only on $q$. This works in general cotangent bundles.