From Do Carmo's book (Riemannian Geometry, P. 19)
If M can be covered by two coordinate neighborhoods $V_1$ and $V_2$ in such a way that the intersection $V_1\cap V_2$ is connected, then $M$ is orientable. Indeed, since the determinant of the differential of the coordinate change is $\neq 0$, it does not change sign in $V_1\cap V_2$ if it is negative at a single point, it suffices to change the sign of one of the coordinates to make it positive at that point, hence on $V_1\cap V_2$.
Why the determinant does not change sign in $V_1\cap V_2$ ? It surely related to the connected assumption, but I miss the argument.