Smooth Manifold, covered by 2 Charts is orientable if the Intersection is Connected

I came across this Question: Atlas on a smooth manifold that contains 2 charts in which Professor Lee commented that this Proposition is true only if the Intersection of the two Maps is connected, so I've been trying to prove the following:

Let $M$ be a smooth Manifold, covered by an Atlas $\mathcal{A}$ containing two charts $\phi,\psi$ and $M= U_\phi \cup V_\psi$. Show that $M$ is orientable if $U_\phi \cap V_\psi$ is connected.

So in other words I will have to prove that,for $\tau = \phi \circ \psi^{-1}$, the following holds: $det(Jac(\tau))>0$.

First of all I tried showing that $det(Jac(\tau)) > 0 \vee det(Jac(\tau)) < 0$ holds. I reasoned : Since $M$ is smooth it follows that $\tau$ is a Diffeomorphism, which implies $\tau$ is continuous. Suppose $\tau$ would take values both bigger and smaller than $0$, this would imply that there is a point at which $\tau$ becomes $0$ and therefore would no longer be a Diffeomorphism. The next step would be to show that $det(Jac(\tau))> 0$ however I do not see why this is indeed the case.

Is my reasoning so far correct and if so how do I continue?

If $\det\bigl(\operatorname{Jac}(\tau)\bigr) < 0$, you can re-orient one of your charts (for example, by precomposing with a linear diffeomorphism that multplies the first coordinate by $-1$).