Two topics in soft collinear effective theory (SCET) for gravitational interactions are explored. First, the collinear Wilson lines---necessary building blocks for maintaining multiple copies of diffeomorphism invariance in gravity SCET---are extended to all orders in the SCET expansion parameter $lambda$, where it has only been known to $O(lambda)$ in the literature. Second, implications of reparametrization invariance (RPI) for the structure of gravity SCET lagrangians are studied. The utility of RPI is illustrated by an explicit example in which $O(lambda^2)$ hard interactions of a collinear graviton are completely predicted by RPI from its $O(lambda)$ hard interactions. It is also pointed out that the multiple diffeomorphism invariances and RPI together require certain relations among $O(lambda)$ terms, thereby reducing the number of $O(lambda)$ terms that need to be fixed by matching onto the full theory in the first place.
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