By Kupka-Smale theorem, a Morse function $f$ on a manifold with a Riemmannian metric $m$ can be perturbed to become Morse-Smale, by genericity. The perturbed function $g$ has identical level-set foliation up to diffeomorphism so it is conjugate by a diffeomorphism $\phi$, namely $g = f \circ \phi$.
The stable and unstable manifolds of $g$ for $m$ are transverse. Next we apply a global change of coordinate to both $g$ and $m$ using $\phi^{-1}$. This will send $g$ back to $f$ and $m$ to its pull-back by $\phi$.
Also this change of coordinate will send the stable/unstable manifolds of $g$ for $m$ to the ones of $f$ for the pull-back of $m$. In particular the new stable/unstable manifolds are transverse. So the pull-back metric satisfies the transversality condition.