In the semiclassical trace formula for the level density of a circular billiard in homogeneous magnetic fields, quantum corrections to the Maslov phase have shown to be important in strong fields. In this article further quantum correcions are considered, namely, grazing corrections, which are relevant for whispering-gallery orbits. and a uniform approximation to the bifurcation points in strong fields is applied. Both corrections are shown to have a surprisingly small effect on the semiclassical level density. Implementing those corrections requires a technique different from the common Gaussian smoothing in the numerical evaluation of the trace formula. The appropiate generalization is presented.