  We consider refinements of the local smoothing estimates for the
Schr\"odinger equation in domains which are exterior to a strictly convex
obstacle in $\RR^n$. By restricting the solution to small, frequency dependent
collars of the boundary, it is expected that taking its square integral in
space-time should exhibit a larger gain in regularity when compared to the
usual gain of half a derivative. By a result of Ivanovici, these refined local
smoothing estimates are satisfied by solutions in the exterior of a ball. We
show that when such estimates are valid, they can be combined with wave packet
parametrix constructions to yield Strichartz estimates. This provides an avenue
for obtaining these bounds when Neumann boundary conditions are imposed.
