We consider the Schramm-Loewner evolution (SLE_{kappa}) for kappa in (4,8), which is the regime that the curve is self-intersecting but not space-filling. We let K be the set of kappa in (4,8) for which the adjacency graph of connected components of the complement of an SLE_{kappa} is almost surely connected, in the sense that for every pair of complementary components U,V there exist complementary components U_1,...,U_n with U_1 = U, U_n = V, and the boundaries of U_i and U_{i+1} intersect for each i between 1 and n-1. We show that the range of an SLE_{kappa} for kappa in K is almost surely conformally removable, which answers a question of Sheffield. As a step in the proof, we construct the canonical conformally covariant volume measure on the cut points of an SLE_{kappa} for kappa in (4,8) and establish a precise upper bound on the measure that it assigns to any Borel set in terms of its diameter. This is joint work with Jason Miller and Lukas Schoug.